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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremequeuclr 2101 Commuted version of equeucl 2102 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑦 = 𝑥))

Theoremequeucl 2102 Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2086.) Curried (exported) form of equtr2 2105. (Contributed by BJ, 11-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))

Theoremequequ1 2103 An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequequ2 2104 An equivalence law for equality. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) (Proof shortened by BJ, 12-Apr-2021.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremequtr2 2105 Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2102. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.)
((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)

Theoremstdpc6 2106 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 2107.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
𝑥 𝑥 = 𝑥

Theoremstdpc7 2107 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 2106.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑥)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
(𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Theoremequvinv 2108* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2164, ax-13 2387. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))

Theoremequviniva 2109* A modified version of the forward implication of equvinv 2108 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))

Theoremequvelv 2110* A specialized version of equvel 2480 with distinct variable restrictions and fewer axiom usage. (Contributed by Wolf Lammen, 10-Apr-2021.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))

Theoremax13b 2111 An equivalence between two ways of expressing ax-13 2387. See the comment for ax-13 2387. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)
((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑))))

Theoremspfw 2112* Weak version of sp 2196. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)

TheoremspfwOLD 2113* Obsolete proof of spfw 2112 as of 10-Oct-2021. (Contributed by NM, 19-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)

Theoremspw 2114* Weak version of the specialization scheme sp 2196. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2196 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2196 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2157 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2196 are spfw 2112 (minimal distinct variable requirements), spnfw 2079 (when 𝑥 is not free in ¬ 𝜑), spvw 2060 (when 𝑥 does not appear in 𝜑), sptruw 1878 (when 𝜑 is true), and spfalw 2080 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)

Theoremcbvalw 2115* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑦𝜓 → ∀𝑥𝑦𝜓)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvalvw 2116* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexvw 2117* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremalcomiw 2118* Weak version of alcom 2182. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremhbn1fw 2119* Weak version of ax-10 2164 from which we can prove any ax-10 2164 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑦𝜓 → ∀𝑥𝑦𝜓)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbn1w 2120* Weak version of hbn1 2165. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhba1w 2121* Weak version of hba1 2294. See comments for ax10w 2151. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhba1wOLD 2122* Obsolete proof of hba1w 2121 as of 10-Oct-2021. (Contributed by NM, 9-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhbe1w 2123* Weak version of hbe1 2166. See comments for ax10w 2151. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhbalw 2124* Weak version of hbal 2181. Uses only Tarski's FOL axiom schemes. Unlike hbal 2181, this theorem requires that 𝑥 and 𝑦 be distinct, i.e. not be bundled. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜑 → ∀𝑥𝜑)       (∀𝑦𝜑 → ∀𝑥𝑦𝜑)

Theoremspaev 2125* A special instance of sp 2196 applied to an equality with a dv condition. Unlike the more general sp 2196, we can prove this without ax-12 2192. Instance of aeveq 2129.

The antecedent 𝑥𝑥 = 𝑦 with distinct 𝑥 and 𝑦 is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term.

Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition 𝑥𝑥 = 𝑦 is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)

Theoremcbvaev 2126* Change bound variable in an equality with a dv condition. Instance of aev 2130. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)

Theoremaevlem0 2127* Lemma for aevlem 2128. Instance of aev 2130. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2192. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 30-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)

Theoremaevlem 2128* Lemma for aev 2130 and axc16g 2277. Change free and bound variables. Instance of aev 2130. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2387, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)

Theoremaeveq 2129* The antecedent 𝑥𝑥 = 𝑦 with a dv condition (typical of a one-object universe) forces equality of everything. (Contributed by Wolf Lammen, 19-Mar-2021.)
(∀𝑥 𝑥 = 𝑦𝑧 = 𝑡)

Theoremaev 2130* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2179. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2387, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2192. (Revised by Wolf Lammen, 19-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢)

Theoremhbaevg 2131* Generalization of hbaev 2132, proved at no extra cost. Instance of aev2 2133. (Contributed by Wolf Lammen, 22-Mar-2021.) (Revised by BJ, 29-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑡 = 𝑢)

Theoremhbaev 2132* Version of hbae 2453 with a DV condition, requiring fewer axioms. Instance of hbaevg 2131 and aev2 2133. (Contributed by Wolf Lammen, 22-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremaev2 2133* A version of aev 2130 with two universal quantifiers in the consequent, and a generalization of hbaevg 2131. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 2129, aev 2130, aev2 2133).

Using aev 2130 and alrimiv 2000 (as in aev2ALT 2134), one can actually prove (with no more axioms) any scheme of the form (∀𝑥𝑥 = 𝑦 PHI) , DV (𝑥, 𝑦) where PHI involves only setvar variables and the connectors , , , , , =, , , ∃*, ∃!, . An example is given by aevdemo 27624. This list cannot be extended to ¬ or since the scheme 𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1867, ax-1 6-- ax-13 2387 (as the one-element universe shows).

(Contributed by BJ, 29-Mar-2021.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)

Theoremaev2ALT 2134* Alternate proof of aev2 2133, bypassing hbaevg 2131. (Contributed by BJ, 23-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)

1.4.9  Membership predicate

Syntaxwcel 2135 Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see mmset.html#class.

(The purpose of introducing wff 𝐴𝐵 here is to allow us to express i.e. "prove" the wel 2136 of predicate calculus in terms of the wcel 2135 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2743 for more information on the set theory usage of wcel 2135.)

wff 𝐴𝐵

Theoremwel 2136 Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read "𝑥 is an element of 𝑦," "𝑥 is a member of 𝑦," "𝑥 belongs to 𝑦," or "𝑦 contains 𝑥." Note: The phrase "𝑦 includes 𝑥 " means "𝑥 is a subset of 𝑦;" to use it also for 𝑥𝑦, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactic construction introduces a binary non-logical predicate symbol (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

(Instead of introducing wel 2136 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 2135. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 2136 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 2135. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

wff 𝑥𝑦

1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)

Axiomax-8 2137 Axiom of Left Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of an arbitrary binary predicate , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be.

We prove in ax8 2141 that this axiom can be recovered from its weakened version ax8v 2138 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-8 2137 should be ax8v 2138. See the comment of ax8v 2138 for more details on these matters. (Contributed by NM, 30-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax8 2141 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremax8v 2138* Weakened version of ax-8 2137, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-8 2137, and it should be referenced only by its two weakened versions ax8v1 2139 and ax8v2 2140, from which ax-8 2137 is then rederived as ax8 2141, which shows that either ax8v 2138 or the conjunction of ax8v1 2139 and ax8v2 2140 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax8 2141 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremax8v1 2139* First of two weakened versions of ax8v 2138, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremax8v2 2140* Second of two weakened versions of ax8v 2138, with an extra dv condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremax8 2141 Proof of ax-8 2137 from ax8v1 2139 and ax8v2 2140, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2138, which is itself a weakened version of ax-8 2137. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremelequ1 2142 An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremcleljust 2143* When the class variables in definition df-clel 2752 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2136 with the class variables in wcel 2135. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2083 in order to remove dependencies on ax-10 2164, ax-12 2192, ax-13 2387. Note that there is no DV condition on 𝑥, 𝑦, that is, on the variables of the left-hand side. (Revised by BJ, 29-Dec-2020.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))

1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)

Axiomax-9 2144 Axiom of Right Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of an arbitrary binary predicate , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint).

We prove in ax9 2148 that this axiom can be recovered from its weakened version ax9v 2145 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-9 2144 should be ax9v 2145. See the comment of ax9v 2145 for more details on these matters. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax9 2148 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax9v 2145* Weakened version of ax-9 2144, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-9 2144, and it should be referenced only by its two weakened versions ax9v1 2146 and ax9v2 2147, from which ax-9 2144 is then rederived as ax9 2148, which shows that either ax9v 2145 or the conjunction of ax9v1 2146 and ax9v2 2147 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax9 2148 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax9v1 2146* First of two weakened versions of ax9v 2145, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax9v2 2147* Second of two weakened versions of ax9v 2145, with an extra dv condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax9 2148 Proof of ax-9 2144 from ax9v1 2146 and ax9v2 2147, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2145, which is itself a weakened version of ax-9 2144. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremelequ2 2149 An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13

The original axiom schemes of Tarski's predicate calculus are ax-4 1882, ax-5 1984, ax6v 2051, ax-7 2086, ax-8 2137, and ax-9 2144, together with rule ax-gen 1867. See mmset.html#compare 1867. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85.

The axiom system of set.mm includes the auxiliary axiom schemes ax-10 2164, ax-11 2179, ax-12 2192, and ax-13 2387, which are not part of Tarski's axiom schemes. Each object-language instance of them is provable from Tarski's axioms, so they are logically redundant. However, they are conjectured not to be provable directly as schemes from Tarski's axiom schemes using only Metamath's direct substitution rule. They are used to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and setvar variables, bundled or not, whose object-language instances are valid. (ax-12 2192 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.)

(There are additional predicate calculus axiom schemes included in set.mm such as ax-c5 34668, but they can all be proved as theorems from the above.)

Terminology: Two setvar (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint (\$d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the 𝑥 and 𝑦 in ax-6 2050 are bundled, but they are not in ax6v 2051. We also say that a scheme is bundled when it has at least one pair of bundled setvar variables. If distinct variable conditions are added to all setvar variable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 2051 is the principal instance of ax-6 2050. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance ¬ ∀𝑥¬ 𝑥 = 𝑥 of ax-6 2050 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions (\$d) to be concerned with, and theorems are more general. There may be some economy in being able to prove facts about principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them).

Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-10 2164, ax-11 2179, ax-12 2192, and ax-13 2387. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 2164, ax-11 2179, ax-12 2192, or ax-13 2387 in which (1) there are no wff metavariables and (2) all setvar variables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each setvar variable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.)

It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object-language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes.

The theorem schemes ax10w 2151, ax11w 2152, ax12w 2155, and ax13w 2158 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 2164, ax-11 2179, ax-12 2192, and ax-13 2387 meeting Conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 2151, ax11w 2152, and ax12w 2155 is of the form (𝑥 = 𝑦 → (𝜑𝜓)) where 𝜓 is an auxiliary or "dummy" wff metavariable in which 𝑥 doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting Conditions (1) and (2). The example ax12wdemo 2157 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this.

We also show the degenerate instances for axioms with bundled variables in ax11dgen 2153, ax12dgen 2156, ax13dgen1 2159, ax13dgen2 2160, ax13dgen3 2161, and ax13dgen4 2162. (Their proofs are trivial, but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-10 2164, ax-11 2179, ax-12 2192, and ax-13 2387 are schemes of Tarski's system, meaning that all object-language instances they generate are theorems of Tarski's system.

It is interesting that Tarski used the bundled scheme ax-6 2050 in an older system, so it seems the main purpose of his later ax6v 2051 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-6 2050 as our official axiom, we show that the degenerate instance holds in ax6dgen 2150. (Recall that in set.mm, the only statement referencing ax-6 2050 is ax6v 2051.)

The case of sp 2196 is curious: originally an axiom scheme of Tarski's system, it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the scheme form 𝑥𝜑𝜑 apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 2114, again requiring substitution instances of 𝜑 that meet Conditions (1) and (2) above. Note that our direct proof sp 2196 requires ax-12 2192, which is not part of Tarski's system.

Theoremax6dgen 2150 Tarski's system uses the weaker ax6v 2051 instead of the bundled ax-6 2050, so here we show that the degenerate case of ax-6 2050 can be derived. Even though ax-6 2050 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 2050 is ax6v 2051. We later rederive from ax6v 2051 the bundled form as ax6 2392 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)
¬ ∀𝑥 ¬ 𝑥 = 𝑥

Theoremax10w 2151* Weak version of ax-10 2164 from which we can prove any ax-10 2164 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. It is an alias of hbn1w 2120 introduced for labeling consistency. (Contributed by NM, 9-Apr-2017.) Use hbn1w 2120 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremax11w 2152* Weak version of ax-11 2179 from which we can prove any ax-11 2179 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2179, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. It is an alias of alcomiw 2118 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomiw 2118 instead. (New usage is discouraged.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremax11dgen 2153 Degenerate instance of ax-11 2179 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
(∀𝑥𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremax12wlem 2154* Lemma for weak version of ax-12 2192. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 2155. (Contributed by NM, 10-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12w 2155* Weak version of ax-12 2192 from which we can prove any ax-12 2192 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that 𝑥 and 𝑦 be distinct (unless 𝑥 does not occur in 𝜑). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for 𝜑, see ax12wdemo 2157. (Contributed by NM, 10-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝑧 → (𝜑𝜒))       (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12dgen 2156 Degenerate instance of ax-12 2192 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
(𝑥 = 𝑥 → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑥𝜑)))

Theoremax12wdemo 2157* Example of an application of ax12w 2155 that results in an instance of ax-12 2192 for a contrived formula with mixed free and bound variables, (𝑥𝑦 ∧ ∀𝑥𝑧𝑥 ∧ ∀𝑦𝑧𝑦𝑥), in place of 𝜑. The proof illustrates bound variable renaming with cbvalvw 2116 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)
(𝑥 = 𝑦 → (∀𝑦(𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥))))

Theoremax13w 2158* Weak version (principal instance) of ax-13 2387. (Because 𝑦 and 𝑧 don't need to be distinct, this actually bundles the principal instance and the degenerate instance 𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥𝑦 = 𝑦)).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 2151, ax11w 2152, and ax12w 2155. (Contributed by NM, 10-Apr-2017.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax13dgen1 2159 Degenerate instance of ax-13 2387 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
𝑥 = 𝑥 → (𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))

Theoremax13dgen2 2160 Degenerate instance of ax-13 2387 where bundled variables 𝑥 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))

Theoremax13dgen3 2161 Degenerate instance of ax-13 2387 where bundled variables 𝑦 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥 𝑦 = 𝑦))

Theoremax13dgen4 2162 Degenerate instance of ax-13 2387 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Oct-2021.)
𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Theoremax13dgen4OLD 2163 Obsolete proof of ax13dgen4 2162 as of 10-Oct-2021. (Contributed by NM, 13-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)

In this section we introduce four additional schemes ax-10 2164, ax-11 2179, ax-12 2192, and ax-13 2387 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "scheme completeness," which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs, and setvar variables ranging over object-language individual variables.

To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax10w 2151, ax11w 2152, ax12w 2155, and ax13w 2158, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all setvar variables to be mutually distinct) can be proved using only the schemes in Tarski's system S2.

An open problem is to show that these four additional schemes are mutually metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 2192 from all others has been shown, and independence of Tarski's ax-6 2050 from all others has been shown; see items 9a and 11 on http://us.metamath.org/award2003.html.

1.5.1  Axiom scheme ax-10 (Quantified Negation)

Axiomax-10 2164 Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2151) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 21-May-2008.) Use its alias hbn1 2165 instead if you must use it. Any theorem in first order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 2164 through ax-13 2387, by invoking ax10w 2151 through ax13w 2158. We encourage proving theorems *without* ax-10 2164 through ax-13 2387 and moving them up to the ax-4 1882 through ax-9 2144 section. (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbn1 2165 Alias for ax-10 2164 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbe1 2166 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by NM, 24-Jan-1993.)
(∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhbe1a 2167 Dual statement of hbe1 2166. Modified version of axc7e 2276 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Theoremnf5-1 2168 One direction of nf5 2259 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Theoremnf5i 2169 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → ∀𝑥𝜑)       𝑥𝜑

Theoremnf5dv 2170* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1855 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnf5dh 2171 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) df-nf 1855 changed. (Revised by Wolf Lammen, 11-Oct-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnfe1 2172 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝜑

Theoremnfa1 2173 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1855 changed. (Revised by Wolf Lammen, 11-Sep-2021.) Remove dependency on ax-12 2192. (Revised by Wolf Lammen, 12-Oct-2021.)
𝑥𝑥𝜑

Theoremnfna1 2174 A convenience theorem particularly designed to remove dependencies on ax-11 2179 in conjunction with distinctors. (Contributed by Wolf Lammen, 2-Sep-2018.)
𝑥 ¬ ∀𝑥𝜑

Theoremnfia1 2175 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥(∀𝑥𝜑 → ∀𝑥𝜓)

Theoremnfnf1 2176 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2192. (Revised by Wolf Lammen, 12-Oct-2021.)
𝑥𝑥𝜑

Theoremmodal-5 2177 The analogue in our predicate calculus of axiom (5) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
(¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Theoremnfe1OLD 2178 Obsolete proof of nfe1 2172 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑

1.5.2  Axiom scheme ax-11 (Quantifier Commutation)

Axiomax-11 2179 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 2152) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremalcoms 2180 Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
(∀𝑥𝑦𝜑𝜓)       (∀𝑦𝑥𝜑𝜓)

Theoremhbal 2181 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by NM, 12-Mar-1993.)
(𝜑 → ∀𝑥𝜑)       (∀𝑦𝜑 → ∀𝑥𝑦𝜑)

Theoremalcom 2182 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
(∀𝑥𝑦𝜑 ↔ ∀𝑦𝑥𝜑)

Theoremalrot3 2183 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦𝑧𝜑 ↔ ∀𝑦𝑧𝑥𝜑)

Theoremalrot4 2184 Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
(∀𝑥𝑦𝑧𝑤𝜑 ↔ ∀𝑧𝑤𝑥𝑦𝜑)

Theoremnfa2 2185 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2192. (Revised by Wolf Lammen, 18-Oct-2021.)
𝑥𝑦𝑥𝜑

Theoremhbald 2186 Deduction form of bound-variable hypothesis builder hbal 2181. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))

Theoremexcom 2187 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-5 1984, ax-6 2050, ax-7 2086, ax-10 2164, ax-12 2192. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
(∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)

Theoremexcomim 2188 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-5 1984, ax-6 2050, ax-7 2086, ax-10 2164, ax-12 2192. (Revised by Wolf Lammen, 8-Jan-2018.)
(∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)

Theoremexcom13 2189 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)

Theoremexrot3 2190 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)

Theoremexrot4 2191 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝑥𝑦𝜑)

1.5.3  Axiom scheme ax-12 (Substitution)

Axiomax-12 2192 Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent 𝑥(𝑥 = 𝑦𝜑) is a way of expressing "𝑦 substituted for 𝑥 in wff 𝜑 " (cf. sb6 2562). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

The original version of this axiom was ax-c15 34674 and was replaced with this shorter ax-12 2192 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2444. Conversely, this axiom is proved from ax-c15 34674 as theorem ax12 2445.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 34674) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 2193 and ax12v2 2194 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2155) but is used as an auxiliary axiom scheme to achieve scheme completeness. (Contributed by NM, 22-Jan-2007.) (New usage is discouraged.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12v 2193* This is essentially axiom ax-12 2192 weakened by additional restrictions on variables. Besides axc11r 2328, this theorem should be the only one referencing ax-12 2192 directly.

Both restrictions on variables have their own value. If for a moment we assume 𝑦 could be set to 𝑥, then, after elimination of the tautology 𝑥 = 𝑥, immediately we have 𝜑 → ∀𝑥𝜑 for all 𝜑 and 𝑥, that is ax-5 1984, a degenerate result.

The second restriction is not necessary, but a simplification that makes the following interpretation easier to see. Since 𝜑 textually at most depends on 𝑥, we can look at it at some given 'fixed' 𝑦. This theorem now states that the truth value of 𝜑 will stay constant, as long as we 'vary 𝑥 around 𝑦' only such that 𝑥 = 𝑦 still holds. Or in other words, equality is the finest grained logical expression. If you cannot differ two sets by =, you won't find a whatever sophisticated expression that does. One might wonder how the described variation of 𝑥 is possible at all. Note that Metamath is a text processor that easily sees a difference between text chunks {𝑥 ∣ ¬ 𝑥 = 𝑥} and {𝑦 ∣ ¬ 𝑦 = 𝑦}. Our usual interpretation is to abstract from textual variations of the same set, but we are free to interpret Metamath's formalism differently, and in fact let 𝑥 run through all textual representations of sets.

Had we allowed 𝜑 to depend also on 𝑦, this idea is both harder to see, and it is less clear that this extra freedom introduces effects not covered by other axioms. (Contributed by Wolf Lammen, 8-Aug-2020.)

(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12v2 2194* It is possible to remove any restriction on 𝜑 in ax12v 2193. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2193 instead when sufficient. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2164 and ax-13 2387. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorem19.8a 2195 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 2057 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2196. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
(𝜑 → ∃𝑥𝜑)

Theoremsp 2196 Specialization. A universally quantified wff implies the wff without a quantifier Axiom scheme B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Also appears as Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). This corresponds to the axiom (T) of modal logic.

For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2486.

This theorem shows that our obsolete axiom ax-c5 34668 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114.

It appears that this scheme cannot be derived directly from Tarski's axioms without auxiliary axiom scheme ax-12 2192. It is thought the best we can do using only Tarski's axioms is spw 2114. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)

(∀𝑥𝜑𝜑)

Theoremspi 2197 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
𝑥𝜑       𝜑

Theoremsps 2198 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem2sp 2199 A double specialization (see sp 2196). Another double specialization, closer to PM*11.1, is 2stdpc4 2487. (Contributed by BJ, 15-Sep-2018.)
(∀𝑥𝑦𝜑𝜑)

Theoremspsd 2200 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

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