P. 21 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	elequ2 2001	An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
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 1734, ax-5 1836, ax6v 1886, ax-7 1932, ax-8 1989, and ax-9 1996, together with rule ax-gen 1719. See mmset.html#compare 1719. 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 2016, ax-11 2031, ax-12 2044, and ax-13 2245, 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 2044 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 33687, 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 ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the 𝑥 and 𝑦 in ax-6 1885 are bundled, but they are not in ax6v 1886. We also say that a scheme is bundled when it has at least one pair of bundled setvar variables. If distinct variable conditions are added to all setvar variable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1886 is the principal instance of ax-6 1885. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance ¬ ∀𝑥¬ 𝑥 = 𝑥 of ax-6 1885 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($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 2016, ax-11 2031, ax-12 2044, and ax-13 2245. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 2016, ax-11 2031, ax-12 2044, or ax-13 2245 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 2003, ax11w 2004, ax12w 2007, and ax13w 2010 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 2016, ax-11 2031, ax-12 2044, and ax-13 2245 meeting Conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 2003, ax11w 2004, and ax12w 2007 is of the form (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) 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 2009 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 2005, ax12dgen 2008, ax13dgen1 2011, ax13dgen2 2012, ax13dgen3 2013, and ax13dgen4 2014. (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 2016, ax-11 2031, ax-12 2044, and ax-13 2245 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 1885 in an older system, so it seems the main purpose of his later ax6v 1886 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 1885 as our official axiom, we show that the degenerate instance holds in ax6dgen 2002. (Recall that in set.mm, the only statement referencing ax-6 1885 is ax6v 1886.) The case of sp 2051 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 ∀𝑥𝜑 → 𝜑 apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 1964, again requiring substitution instances of 𝜑 that meet Conditions (1) and (2) above. Note that our direct proof sp 2051 requires ax-12 2044, which is not part of Tarski's system.
Theorem	ax6dgen 2002	Tarski's system uses the weaker ax6v 1886 instead of the bundled ax-6 1885, so here we show that the degenerate case of ax-6 1885 can be derived. Even though ax-6 1885 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1885 is ax6v 1886. We later rederive from ax6v 1886 the bundled form as ax6 2250 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥
Theorem	ax10w 2003*	Weak version of ax-10 2016 from which we can prove any ax-10 2016 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	ax11w 2004*	Weak version of ax-11 2031 from which we can prove any ax-11 2031 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2031, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	ax11dgen 2005	Degenerate instance of ax-11 2031 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
⊢ (∀𝑥∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	ax12wlem 2006*	Lemma for weak version of ax-12 2044. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 2007. (Contributed by NM, 10-Apr-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12w 2007*	Weak version of ax-12 2044 from which we can prove any ax-12 2044 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 2009. (Contributed by NM, 10-Apr-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12dgen 2008	Degenerate instance of ax-12 2044 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 = 𝑥 → 𝜑)))
Theorem	ax12wdemo 2009*	Example of an application of ax12w 2007 that results in an instance of ax-12 2044 for a contrived formula with mixed free and bound variables, (𝑥 ∈ 𝑦 ∧ ∀𝑥𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧𝑦 ∈ 𝑥), in place of 𝜑. The proof illustrates bound variable renaming with cbvalvw 1966 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(𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥))))
Theorem	ax13w 2010*	Weak version (principal instance) of ax-13 2245. (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 2003, ax11w 2004, and ax12w 2007. (Contributed by NM, 10-Apr-2017.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax13dgen1 2011	Degenerate instance of ax-13 2245 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))
Theorem	ax13dgen2 2012	Degenerate instance of ax-13 2245 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))
Theorem	ax13dgen3 2013	Degenerate instance of ax-13 2245 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))
Theorem	ax13dgen4 2014	Degenerate instance of ax-13 2245 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))
Theorem	ax13dgen4OLD 2015	Obsolete proof of ax13dgen4 2014 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 2016, ax-11 2031, ax-12 2044, and ax-13 2245 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 2003, ax11w 2004, ax12w 2007, and ax13w 2010, 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 2044 from all others has been shown, and independence of Tarski's ax-6 1885 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)
Axiom	ax-10 2016	Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2003) 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 2017 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 2016 through ax-13 2245, by invoking ax10w 2003 through ax13w 2010. We encourage proving theorems *without* ax-10 2016 through ax-13 2245 and moving them up to the ax-4 1734 through ax-9 1996 section. (New usage is discouraged.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	hbn1 2017	Alias for ax-10 2016 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	hbe1 2018	The setvar 𝑥 is not free in ∃𝑥𝜑. (Contributed by NM, 24-Jan-1993.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	hbe1a 2019	Dual statement of hbe1 2018. Modified version of axc7e 2129 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
Theorem	nf5-1 2020	One direction of nf5 2113 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
Theorem	nf5i 2021	Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nf5dv 2022*	Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nf5dh 2023	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 11-Oct-2021.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nfe1 2024	The setvar 𝑥 is not free in ∃𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥∃𝑥𝜑
Theorem	nfa1 2025	The setvar 𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 11-Sep-2021.) Remove dependency on ax-12 2044. (Revised by Wolf Lammen, 12-Oct-2021.)
⊢ Ⅎ𝑥∀𝑥𝜑
Theorem	nfna1 2026	A convenience theorem particularly designed to remove dependencies on ax-11 2031 in conjunction with distinctors. (Contributed by Wolf Lammen, 2-Sep-2018.)
⊢ Ⅎ𝑥 ¬ ∀𝑥𝜑
Theorem	nfia1 2027	Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥(∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	nfnf1 2028	The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2044. (Revised by Wolf Lammen, 12-Oct-2021.)
⊢ Ⅎ𝑥Ⅎ𝑥𝜑
Theorem	modal-5 2029	The analogue in our predicate calculus of axiom (5) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
Theorem	nfe1OLD 2030	Obsolete proof of nfe1 2024 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥∃𝑥𝜑
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)
Axiom	ax-11 2031	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 2004) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	alcoms 2032	Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
⊢ (∀𝑥∀𝑦𝜑 → 𝜓)    ⇒   ⊢ (∀𝑦∀𝑥𝜑 → 𝜓)
Theorem	hbal 2033	If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
Theorem	alcom 2034	Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)
Theorem	alrot3 2035	Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑦∀𝑧∀𝑥𝜑)
Theorem	alrot4 2036	Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)
Theorem	nfa2 2037	Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2044. (Revised by Wolf Lammen, 18-Oct-2021.)
⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑
Theorem	hbald 2038	Deduction form of bound-variable hypothesis builder hbal 2033. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))
Theorem	excom 2039	Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-5 1836, ax-6 1885, ax-7 1932, ax-10 2016, ax-12 2044. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
Theorem	excomim 2040	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 1836, ax-6 1885, ax-7 1932, ax-10 2016, ax-12 2044. (Revised by Wolf Lammen, 8-Jan-2018.)
⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
Theorem	excom13 2041	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑)
Theorem	exrot3 2042	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑)
Theorem	exrot4 2043	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)
1.5.3  Axiom scheme ax-12 (Substitution)
Axiom	ax-12 2044	Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent ∀𝑥(𝑥 = 𝑦 → 𝜑) is a way of expressing "𝑦 substituted for 𝑥 in wff 𝜑 " (cf. sb6 2428). 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 33693 and was replaced with this shorter ax-12 2044 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2302. Conversely, this axiom is proved from ax-c15 33693 as theorem ax12 2303. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 33693) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. See ax12v 2045 and ax12v2 2046 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax12w 2007) 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 = 𝑦 → 𝜑)))
Theorem	ax12v 2045*	This is essentially axiom ax-12 2044 weakened by additional restrictions on variables. Besides axc11r 2186, this theorem should be the only one referencing ax-12 2044 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 1836, 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 = 𝑦 → 𝜑)))
Theorem	ax12v2 2046*	It is possible to remove any restriction on 𝜑 in ax12v 2045. Same as Axiom C8 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2016 and ax-13 2245. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12vOLD 2047*	Obsolete proof of ax12v2 2046 as of 24-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2016 and ax-13 2245. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof shortened by Wolf Lammen, 7-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12vOLDOLD 2048*	Obsolete proof of ax12v 2045 as of 7-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2016 and ax-13 2245. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	19.8a 2049	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 1892 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2051. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
⊢ (𝜑 → ∃𝑥𝜑)
Theorem	19.8aOLD 2050	Obsolete proof of 19.8a 2049. Obsolete as of 21-Dec-2020. Can be deleted as soon as the question of why "MM-PA> min exlimiiv" does not give 19.8a 2049 is answered. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2051. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥𝜑)
Theorem	sp 2051	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 2352. This theorem shows that our obsolete axiom ax-c5 33687 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 2044. It is thought the best we can do using only Tarski's axioms is spw 1964. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	spi 2052	Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
⊢ ∀𝑥𝜑    ⇒   ⊢ 𝜑
Theorem	sps 2053	Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	2sp 2054	A double specialization (see sp 2051). Another double specialization, closer to PM*11.1, is 2stdpc4 2353. (Contributed by BJ, 15-Sep-2018.)
⊢ (∀𝑥∀𝑦𝜑 → 𝜑)
Theorem	spsd 2055	Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	19.2g 2056	Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1889 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
⊢ (∀𝑥𝜑 → ∃𝑦𝜑)
Theorem	19.21bi 2057	Inference form of 19.21 2073 and also deduction form of sp 2051. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.21bbi 2058	Inference removing double quantifier. Version of 19.21bi 2057 with two quanditiers. (Contributed by NM, 20-Apr-1994.)
⊢ (𝜑 → ∀𝑥∀𝑦𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.23bi 2059	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2078. (Contributed by NM, 12-Mar-1993.)
⊢ (∃𝑥𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	nexr 2060	Inference associated with the contrapositive of 19.8a 2049. (Contributed by Jeff Hankins, 26-Jul-2009.)
⊢ ¬ ∃𝑥𝜑    ⇒   ⊢ ¬ 𝜑
Theorem	qexmid 2061	Quantified excluded middle (see exmid 431). Also known as the drinker paradox (if 𝜑(𝑥) is interpreted as "𝑥 drinks", then this theorem tells that there exists a person such that, if this person drinks, then everyone drinks). Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)
⊢ ∃𝑥(𝜑 → ∀𝑥𝜑)
Theorem	nf5r 2062	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Theorem	nf5ri 2063	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	nf5rd 2064	Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	nfim1 2065	A closed form of nfim 1822. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1707 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜑 → 𝜓)
Theorem	nfan1 2066	A closed form of nfan 1825. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
Theorem	19.3 2067	A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1894 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥𝜑 ↔ 𝜑)
Theorem	19.9d 2068	A deduction version of one direction of 19.9 2070. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1707 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
⊢ (𝜓 → Ⅎ𝑥𝜑)    ⇒   ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))
Theorem	19.9t 2069	A closed version of 19.9 2070. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
Theorem	19.9 2070	A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1893 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	19.21t 2071	Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2073. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1707 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	19.21tOLDOLD 2072	Obsolete proof of 19.21t 2071 as of 3-Nov-2021. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1707 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	19.21 2073	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See 19.21v 1865 for a version requiring fewer axioms. See also 19.21h 2118. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	stdpc5 2074	An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎ𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 1937. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1864 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2016. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	stdpc5OLD 2075	Obsolete proof of stdpc5 2074 as of 11-Oct-2021. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2016. (Revised by Wolf Lammen, 4-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	19.21-2 2076	Version of 19.21 2073 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	19.23t 2077	Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2078. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1707 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	19.23 2078	Theorem 19.23 of [Margaris] p. 90. See 19.23v 1899 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	alimd 2079	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1735. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	alrimi 2080	Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2073. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimdd 2081	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2073. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	alrimd 2082	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2073. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	eximd 2083	Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1758. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	exlimi 2084	Inference associated with 19.23 2078. See exlimiv 1855 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimd 2085	Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimdd 2086	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	nexd 2087	Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥𝜓)
Theorem	albid 2088	Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbid 2089	Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	nfbidf 2090	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1707 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	19.16 2091	Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓))
Theorem	19.17 2092	Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))
Theorem	19.27 2093	Theorem 19.27 of [Margaris] p. 90. See 19.27v 1905 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.28 2094	Theorem 19.28 of [Margaris] p. 90. See 19.28v 1906 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Theorem	19.19 2095	Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓))
Theorem	19.36 2096	Theorem 19.36 of [Margaris] p. 90. See 19.36v 1901 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
Theorem	19.36i 2097	Inference associated with 19.36 2096. See 19.36iv 1902 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
⊢ Ⅎ𝑥𝜓    &   ⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.37 2098	Theorem 19.37 of [Margaris] p. 90. See 19.37v 1907 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))
Theorem	19.32 2099	Theorem 19.32 of [Margaris] p. 90. See 19.32v 1866 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Theorem	19.31 2100	Theorem 19.31 of [Margaris] p. 90. See 19.31v 1867 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))