Home Metamath Proof ExplorerTheorem List (p. 21 of 431) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28080) Hilbert Space Explorer (28081-29605) Users' Mathboxes (29606-43060)

Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremalrimivv 2001* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2218 and 19.21v 2013. (Contributed by NM, 31-Jul-1995.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝑦𝜓)

Theoremalrimdv 2002* Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2218 and 19.21v 2013. (Contributed by NM, 10-Feb-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremexlimiv 2003* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2223.

See exlimi 2229 for a more general version requiring more axioms.

This inference, along with its many variants such as rexlimdv 3164, is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.appstate.edu/~hirstjl/primer/hirst.pdf. In informal proofs, the statement "Let 𝐶 be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element 𝑥 exists satisfying a wff, i.e. 𝑥𝜑(𝑥) where 𝜑(𝑥) has 𝑥 free, then we can use 𝜑(𝐶) as a hypothesis for the proof where 𝐶 is a new (fictitious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original 𝜑 (containing 𝑥) as an antecedent for the main part of the proof. We eventually arrive at (𝜑𝜓) where 𝜓 is the theorem to be proved and does not contain 𝑥. Then we apply exlimiv 2003 to arrive at (∃𝑥𝜑𝜓). Finally, we separately prove 𝑥𝜑 and detach it with modus ponens ax-mp 5 to arrive at the final theorem 𝜓. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 2050 and ax-8 2137. (Revised by Wolf Lammen, 4-Dec-2017.)

(𝜑𝜓)       (∃𝑥𝜑𝜓)

Theoremexlimiiv 2004* Inference associated with exlimiv 2003. (Contributed by BJ, 19-Dec-2020.)
(𝜑𝜓)    &   𝑥𝜑       𝜓

Theoremexlimivv 2005* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2223. (Contributed by NM, 1-Aug-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑𝜓)

Theoremexlimdv 2006* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2223. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 2050, ax-7 2086. (Revised by Wolf Lammen, 4-Dec-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

Theoremexlimdvv 2007* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2223. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓𝜒))

Theoremexlimddv 2008* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theoremnexdv 2009* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

TheoremnexdvOLD 2010* Obsolete proof of nexdv 2009 as of 10-Oct-2021. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theorem2ax5 2011* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(𝜑 → ∀𝑥𝑦𝜑)

Theoremstdpc5v 2012* Version of stdpc5 2219 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 2013. (Revised by Wolf Lammen, 12-Jul-2020.)
(∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem19.21v 2013* Version of 19.21 2218 with a dv condition, requiring fewer axioms.

Notational convention: We sometimes suffix with "v" the label of a theorem using a distinct variable ("dv") condition instead of a non-freeness hypothesis such as 𝑥𝜑. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a non-freeness hypothesis ("f" stands for "not free in", see df-nf 1855) instead of a dv condition. For instance, 19.21v 2013 versus 19.21 2218 and vtoclf 3394 versus vtocl 3395. Note that "not free in" is less restrictive than "does not occur in." Note that the version with a dv condition is easily proved from the version with the corresponding non-freeness hypothesis, by using nfv 1988. However, the dv version can often be proved from fewer axioms. (Contributed by NM, 21-Jun-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (Proof shortened by Wolf Lammen, 12-Jul-2020.)

(∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theorem19.32v 2014* Version of 19.32 2244 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Theorem19.31v 2015* Version of 19.31 2245 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.23v 2016* Version of 19.23 2223 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 28-Jun-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 11-Jan-2020.) Remove dependency on ax-6 2050. (Revised by Rohan Ridenour, 15-Apr-2022.)
(∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.23vv 2017* Theorem 19.23v 2016 extended to two variables. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theorem19.36imv 2018* One direction of 19.36v 2065 that can be proven without ax-6 2050. (Contributed by Rohan Ridenour, 16-Apr-2022.)
(∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorem19.36iv 2019* Inference associated with 19.36v 2065. Version of 19.36i 2242 with a dv condition. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.) Remove dependency on ax-6 2050. (Revised by Rohan Ridenour, 15-Apr-2022.)
𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem19.37imv 2020* One direction of 19.37v 2071 that can be proven without ax-6 2050. (Contributed by Rohan Ridenour, 16-Apr-2022.)
(∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))

Theorem19.37iv 2021* Inference associated with 19.37v 2071. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 2050. (Revised by Rohan Ridenour, 15-Apr-2022.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.41v 2022* Version of 19.41 2246 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-6 2050. (Revised by Rohan Ridenour, 15-Apr-2022.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.41vv 2023* Version of 19.41 2246 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theorem19.41vvv 2024* Version of 19.41 2246 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))

Theorem19.41vvvv 2025* Version of 19.41 2246 with four quantifiers and a dv condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))

Theorem19.42v 2026* Version of 19.42 2248 with a dv condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theoremexdistr 2027* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theorem19.42vv 2028* Version of 19.42 2248 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝜓))

Theorem19.42vvv 2029* Version of 19.42 2248 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))

Theoremexdistr2 2030* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))

Theorem3exdistr 2031* Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))

Theorem4exdistr 2032* Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
(∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))

TheoremnfvOLD 2033* Obsolete proof of nfv 1988 as of 5-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑

TheoremnfvdOLD 2034* Obsolete proof of nfvd 1989 as of 6-Oct-2021. (Contributed by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)

TheoremnfdvOLD 2035* Obsolete proof of nf5dv 2170 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

1.4.5  Equality predicate (continued)

The equality predicate was introduced above in wceq 1628 for use by df-tru 1631. See the comments in that section. In this section, we continue with the first "real" use of it.

Theoremweq 2036 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 2036 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 wceq 1628. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 2036 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1628. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

wff 𝑥 = 𝑦

Theoremequs3 2037 Lemma used in proofs of substitution properties. (Contributed by NM, 10-May-1993.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))

Theoremspeimfw 2038 Specialization, with additional weakening (compared to 19.2 2054) to allow bundling of 𝑥 and 𝑦. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Dec-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

TheoremspeimfwALT 2039 Alternate proof of speimfw 2038 (longer compressed proof, but fewer essential steps). (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))

Theoremspimfw 2040 Specialization, with additional weakening (compared to sp 2196) to allow bundling of 𝑥 and 𝑦. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑𝜓))

Theoremax12i 2041 Inference that has ax-12 2192 (without 𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2192 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

1.4.6  Define proper substitution

Syntaxwsb 2042 Extend wff definition to include proper substitution (read "the wff that results when 𝑦 is properly substituted for 𝑥 in wff 𝜑"). (Contributed by NM, 24-Jan-2006.)
wff [𝑦 / 𝑥]𝜑

Definitiondf-sb 2043 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [𝑦 / 𝑥]𝜑 to mean "the wff that results from the proper substitution of 𝑦 for 𝑥 in the wff 𝜑." That is, 𝑦 properly replaces 𝑥. For example, [𝑥 / 𝑦]𝑧𝑦 is the same as 𝑧𝑥, as shown in elsb4 2568. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2486.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑦) is the wff that results when 𝑦 is properly substituted for 𝑥 in 𝜑(𝑥)." For example, if the original 𝜑(𝑥) is 𝑥 = 𝑦, then 𝜑(𝑦) is 𝑦 = 𝑦, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2509, sbcom2 2578 and sbid2v 2590).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2257 shows. We achieve this by having 𝑥 free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2588 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2507. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2563 and sb6 2562.

There are no restrictions on any of the variables, including what variables may occur in wff 𝜑. (Contributed by NM, 10-May-1993.)

([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theoremsbequ2 2044 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))

Theoremsb1 2045 One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2563) or a non-freeness hypothesis (sb5f 2519). (Contributed by NM, 13-May-1993.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremspsbe 2046 A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.)
([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Theoremsbequ8 2047 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))

Theoremsbimi 2048 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)

Theoremsbbii 2049 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)

1.4.7  Axiom scheme ax-6 (Existence)

Axiomax-6 2050 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is that at least one thing exists. In this form (not requiring that 𝑥 and 𝑦 be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by axc10 2393 and ax6fromc10 34681. A more convenient form of this axiom is ax6e 2391, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.

ax-6 2050 can be proved from the weaker version ax6v 2051 requiring that the variables be distinct; see theorem ax6 2392.

ax-6 2050 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax6vsep 4933.

Except by ax6v 2051, this axiom should not be referenced directly. Instead, use theorem ax6 2392. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦

Theoremax6v 2051* Axiom B7 of [Tarski] p. 75, which requires that 𝑥 and 𝑦 be distinct. This trivial proof is intended merely to weaken axiom ax-6 2050 by adding a distinct variable restriction (\$d). From here on, ax-6 2050 should not be referenced directly by any other proof, so that theorem ax6 2392 will show that we can recover ax-6 2050 from this weaker version if it were an axiom (as it is in the case of Tarski).

Note: Introducing 𝑥, 𝑦 as a distinct variable group "out of the blue" with no apparent justification has puzzled some people, but it is perfectly sound. All we are doing is adding an additional prerequisite, similar to adding an unnecessary logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax6v 2051 must have a \$d specified for the two variables that get substituted for 𝑥 and 𝑦. The \$d does not propagate "backwards" i.e. it does not impose a requirement on ax-6 2050.

When possible, use of this theorem rather than ax6 2392 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦

Theoremax6ev 2052* At least one individual exists. Weaker version of ax6e 2391. When possible, use of this theorem rather than ax6e 2391 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦

Theoremexiftru 2053 Rule of existential generalization, similar to universal generalization ax-gen 1867, but valid only if an individual exists. Its proof requires ax-6 2050 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1867, ax-4 1882 and this theorem alone, not requiring ax-7 2086 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
𝜑       𝑥𝜑

Theorem19.2 2054 Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 2201 for a more conventional proof of a more general result, which uses additional axioms. The reverse implication is the defining property of effective non-freeness (see df-nf 1855). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2086. (Revised by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑 → ∃𝑥𝜑)

Theorem19.2d 2055 Deduction associated with 19.2 2054. (Contributed by BJ, 12-May-2019.)
(𝜑 → ∀𝑥𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.8w 2056 Weak version of 19.8a 2195 and instance of 19.2d 2055. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)       (𝜑 → ∃𝑥𝜑)

Theorem19.8v 2057* Version of 19.8a 2195 with a dv condition, requiring fewer axioms. Converse of ax5e 1986. (Contributed by BJ, 12-Mar-2020.)
(𝜑 → ∃𝑥𝜑)

Theorem19.9v 2058* Version of 19.9 2215 with a dv condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 2059. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 2086. (Revised by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)

Theorem19.3v 2059* Version of 19.3 2212 with a dv condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 2058. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2086. (Revised by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑𝜑)

Theoremspvw 2060* Version of sp 2196 when 𝑥 does not occur in 𝜑. Converse of ax-5 1984. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑𝜑)

Theorem19.39 2061 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.24 2062 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.34 2063 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.23vOLD 2064* Obsolete version of 19.23v 2016 as of 15-Apr-2022. (Contributed by NM, 28-Jun-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 11-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.36v 2065* Version of 19.36 2241 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
(∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.36ivOLD 2066* Obsolete version of 19.36iv 2019 as of 15-Apr-2022. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorempm11.53v 2067* Version of pm11.53 2320 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))

Theorem19.12vvv 2068* Version of 19.12vv 2321 with a dv condition, requiring fewer axioms. See also 19.12 2305. (Contributed by BJ, 18-Mar-2020.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))

Theorem19.27v 2069* Version of 19.27 2238 with a dv condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28v 2070* Version of 19.28 2239 with a dv condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.37v 2071* Version of 19.37 2243 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Theorem19.37ivOLD 2072* Obsolete version of 19.37iv 2021 as of 15-Apr-2022. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.44v 2073* Version of 19.44 2249 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45v 2074* Version of 19.45 2250 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theorem19.41vOLD 2075* Obsolete version of 19.41v 2022 as of 15-Apr-2022. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theoremspimeh 2076* Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
(𝜑 → ∀𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspimw 2077* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimvw 2078* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspnfw 2079 Weak version of sp 2196. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)
𝜑 → ∀𝑥 ¬ 𝜑)       (∀𝑥𝜑𝜑)

Theoremspfalw 2080 Version of sp 2196 when 𝜑 is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
¬ 𝜑       (∀𝑥𝜑𝜑)

Theoremequs4v 2081* Version of equs4 2431 with a dv condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremequsalvw 2082* Version of equsalv 2251 with a dv condition, and of equsal 2432 with two dv conditions, which requires fewer axioms. See also the dual form equsexvw 2083. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexvw 2083* Version of equsexv 2252 with a dv condition, and of equsex 2433 with two dv conditions, which requires fewer axioms. See also the dual form equsalvw 2082. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremcbvaliw 2084* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbvalivw 2085* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

1.4.8  Axiom scheme ax-7 (Equality)

Axiomax-7 2086 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. It states that equality is a right-Euclidean binary relation (this is similar, but not identical, to being transitive, which is proved as equtr 2099). 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 C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1557 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

We prove in ax7 2094 that this axiom can be recovered from its weakened version ax7v 2087 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 2086 should be ax7v 2087. See the comment of ax7v 2087 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 2094 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v 2087* Weakened version of ax-7 2086, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2086, and it should be referenced only by its two weakened versions ax7v1 2088 and ax7v2 2089, from which ax-7 2086 is then rederived as ax7 2094, which shows that either ax7v 2087 or the conjunction of ax7v1 2088 and ax7v2 2089 is sufficient.

In ax7v 2087, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2087 "bundles" (a term coined by Raph Levien) its "principal instance" (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧)) with 𝑥, 𝑦, 𝑧 distinct, and its "degenerate instances" (𝑥 = 𝑦 → (𝑥 = 𝑥𝑦 = 𝑥)) and (𝑥 = 𝑦 → (𝑥 = 𝑦𝑦 = 𝑦)) with 𝑥, 𝑦 distinct. These degenerate instances are for instance used in the proofs of equcomiv 2092 and equid 2090 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2094 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v1 2088* First of two weakened versions of ax7v 2087, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v2 2089* Second of two weakened versions of ax7v 2087, with an extra dv condition on 𝑦, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequid 2090 Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
𝑥 = 𝑥

Theoremnfequid 2091 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
𝑦 𝑥 = 𝑥

Theoremequcomiv 2092* Weaker form of equcomi 2095 with a dv condition on 𝑥, 𝑦. This is an intermediate step and equcomi 2095 is fully recovered later. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremax6evr 2093* A commuted form of ax6ev 2052. (Contributed by BJ, 7-Dec-2020.)
𝑥 𝑦 = 𝑥

Theoremax7 2094 Proof of ax-7 2086 from ax7v1 2088 and ax7v2 2089, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2087, which is itself a weakened version of ax-7 2086.

Note that the weakened version of ax-7 2086 obtained by adding a dv condition on 𝑥, 𝑧 (resp. on 𝑦, 𝑧) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequcomi 2095 Commutative law for equality. Equality is a symmetric relation. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcom 2096 Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcomd 2097 Deduction form of equcom 2096, symmetry of equality. For the versions for classes, see eqcom 2763 and eqcomd 2762. (Contributed by BJ, 6-Oct-2019.)
(𝜑𝑥 = 𝑦)       (𝜑𝑦 = 𝑥)

Theoremequcoms 2098 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.)
(𝑥 = 𝑦𝜑)       (𝑦 = 𝑥𝜑)

Theoremequtr 2099 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))

Theoremequtrr 2100 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43060
 Copyright terms: Public domain < Previous  Next >