P. 24 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hbim1 2301	A closed form of hbim 2303. (Contributed by NM, 2-Jun-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	hbimd 2302	Deduction form of bound-variable hypothesis builder hbim 2303. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))
Theorem	hbim 2303	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	hban 2304	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
Theorem	hb3an 2305	If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	sbi2 2306	Introduction of implication into substitution. (Contributed by NM, 14-May-1993.)
⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
Theorem	sbim 2307	Implication inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sbrim 2308	Substitution in an implication with a variable not free in the antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) Avoid ax-10 2141. (Revised by GG, 20-Nov-2024.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sbrimOLD 2309	Obsolete version of sbrim 2308 as of 20-Nov-2024. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sblim 2310	Substitution in an implication with a variable not free in the consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))
Theorem	sbor 2311	Disjunction inside and outside of a substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
Theorem	sbbi 2312	Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.)
⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Theorem	sblbis 2313	Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥](𝜒 ↔ 𝜑) ↔ ([𝑦 / 𝑥]𝜒 ↔ 𝜓))
Theorem	sbrbis 2314	Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ [𝑦 / 𝑥]𝜒))
Theorem	sbrbif 2315	Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
⊢ Ⅎ𝑥𝜒    &   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))
Theorem	sbnf 2316*	Move nonfree predicate in and out of substitution; see sbal 2170 and sbex 2285. (Contributed by BJ, 2-May-2019.) (Proof shortened by Wolf Lammen, 2-May-2025.)
⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
Theorem	sbnfOLD 2317*	Obsolete version of sbnf 2316 as of 2-May-2025. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
Theorem	sbiev 2318*	Conversion of implicit substitution to explicit substitution. Version of sbie 2510 with a disjoint variable condition, not requiring ax-13 2380. See sbievw 2093 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2141 and shorten proof. (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Jul-2025.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbievOLD 2319*	Obsolete version of sbiev 2318 as of 24-Aug-2025. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2141 and shorten proof. (Revised by BJ, 18-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbiedw 2320*	Conversion of implicit substitution to explicit substitution (deduction version of sbiev 2318). Version of sbied 2511 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	axc7 2321	Show that the original axiom ax-c7 38841 can be derived from ax-10 2141 (hbn1 2142), sp 2184 and propositional calculus. See ax10fromc7 38851 for the rederivation of ax-10 2141 from ax-c7 38841. Normally, axc7 2321 should be used rather than ax-c7 38841, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc7e 2322	Abbreviated version of axc7 2321 using the existential quantifier. Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
⊢ (∃𝑥∀𝑥𝜑 → 𝜑)
Theorem	modal-b 2323	The analogue in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
Theorem	19.9ht 2324	A closed version of 19.9h 2290. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
Theorem	axc4 2325	Show that the original axiom ax-c4 38840 can be derived from ax-4 1807 (alim 1808), ax-10 2141 (hbn1 2142), sp 2184 and propositional calculus. See ax4fromc4 38850 for the rederivation of ax-4 1807 from ax-c4 38840. Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	axc4i 2326	Inference version of axc4 2325. (Contributed by NM, 3-Jan-1993.)
⊢ (∀𝑥𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	nfal 2327	If 𝑥 is not free in 𝜑, then it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∀𝑦𝜑
Theorem	nfex 2328	If 𝑥 is not free in 𝜑, then it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Reduce symbol count in nfex 2328, hbex 2329. (Revised by Wolf Lammen, 16-Oct-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃𝑦𝜑
Theorem	hbex 2329	If 𝑥 is not free in 𝜑, then it is not free in ∃𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2328, hbex 2329. (Revised by Wolf Lammen, 16-Oct-2021.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
Theorem	nfnf 2330	If 𝑥 is not free in 𝜑, then it is not free in Ⅎ𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥Ⅎ𝑦𝜑
Theorem	19.12 2331	Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 2353 and r19.12sn 4745. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)
Theorem	nfald 2332	Deduction form of nfal 2327. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 16-Oct-2021.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	nfexd 2333	If 𝑥 is not free in 𝜓, then it is not free in ∃𝑦𝜓. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
Theorem	nfsbv 2334*	If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is disjoint from both 𝑥 and 𝑦. Version of nfsb 2531 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2380. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
Theorem	nfsbvOLD 2335*	Obsolete version of nfsbv 2334 as of 25-Oct-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
Theorem	sbco2v 2336*	A composition law for substitution. Version of sbco2 2519 with disjoint variable conditions but not requiring ax-13 2380. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 29-Apr-2023.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	aaan 2337	Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2141. (Revised by GG, 21-Nov-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Theorem	aaanOLD 2338	Obsolete version of aaan 2337 as of 21-Nov-2024. (Contributed by NM, 12-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Theorem	eeor 2339	Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2141. (Revised by GG, 21-Nov-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
Theorem	eeorOLD 2340	Obsolete version of eeor 2339 as of 21-Nov-2024. (Contributed by NM, 8-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
Theorem	cbv3v 2341*	Rule used to change bound variables, using implicit substitution. Version of cbv3 2405 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	cbv1v 2342*	Rule used to change bound variables, using implicit substitution. Version of cbv1 2410 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	cbv2w 2343*	Rule used to change bound variables, using implicit substitution. Version of cbv2 2411 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvaldw 2344*	Deduction used to change bound variables, using implicit substitution. Version of cbvald 2415 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvexdw 2345*	Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2416 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	cbv3hv 2346*	Rule used to change bound variables, using implicit substitution. Version of cbv3h 2412 with a disjoint variable condition on 𝑥, 𝑦, which does not require ax-13 2380. Was used in a proof of axc11n 2434 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	cbvalv1 2347*	Rule used to change bound variables, using implicit substitution. Version of cbval 2406 with a disjoint variable condition, which does not require ax-13 2380. See cbvalvw 2035 for a version with two more disjoint variable conditions, requiring fewer axioms, and cbvalv 2408 for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	cbvexv1 2348*	Rule used to change bound variables, using implicit substitution. Version of cbvex 2407 with a disjoint variable condition, which does not require ax-13 2380. See cbvexvw 2036 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2409 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	cbval2v 2349*	Rule used to change bound variables, using implicit substitution. Version of cbval2 2419 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 22-Dec-2003.) (Revised by BJ, 16-Jun-2019.) (Proof shortened by GG, 10-Jan-2024.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	cbvex2v 2350*	Rule used to change bound variables, using implicit substitution. Version of cbvex2 2420 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 14-Sep-2003.) (Revised by BJ, 16-Jun-2019.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	dvelimhw 2351*	Proof of dvelimh 2458 without using ax-13 2380 but with additional distinct variable conditions. (Contributed by NM, 1-Oct-2002.) (Revised by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 23-Dec-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑧𝜓)    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Theorem	pm11.53 2352*	Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1943 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Theorem	19.12vv 2353*	Special case of 19.12 2331 where its converse holds. See 19.12vvv 1988 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
Theorem	eean 2354	Distribute existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	eeanv 2355*	Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1949 and 19.42v 1953. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1955. (Contributed by NM, 26-Jul-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	eeeanv 2356*	Distribute three existential quantifiers over a conjunction. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.)
⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
Theorem	ee4anv 2357*	Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv 1956. (Contributed by NM, 31-Jul-1995.)
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
Theorem	sb8v 2358*	Substitution of variable in universal quantifier. Version of sb8f 2359 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2178. (Contributed by SN, 5-Dec-2024.)
⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb8f 2359*	Substitution of variable in universal quantifier. Version of sb8 2525 with a disjoint variable condition, not requiring ax-10 2141 or ax-13 2380. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2141. (Revised by SN, 5-Dec-2024.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb8fOLD 2360*	Obsolete version of sb8f 2359 as of 5-Dec-2024. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb8ef 2361*	Substitution of variable in existential quantifier. Version of sb8e 2526 with a disjoint variable condition, not requiring ax-13 2380. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Theorem	2sb8ef 2362*	An equivalent expression for double existence. Version of 2sb8e 2538 with more disjoint variable conditions, not requiring ax-13 2380. (Contributed by Wolf Lammen, 28-Jan-2023.)
⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Theorem	sb6rfv 2363*	Reversed substitution. Version of sb6rf 2476 requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993.) (Revised by Wolf Lammen, 7-Feb-2023.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
Theorem	sbnf2 2364*	Two ways of expressing "𝑥 is (effectively) not free in 𝜑". (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.) Avoid ax-13 2380. (Revised by Wolf Lammen, 30-Jan-2023.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
Theorem	exsb 2365*	An equivalent expression for existence. One direction (exsbim 2001) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2380. (Revised by Wolf Lammen, 16-Oct-2022.)
⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	2exsb 2366*	An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
Theorem	sbbib 2367*	Reversal of substitution. (Contributed by AV, 6-Aug-2023.) (Proof shortened by Wolf Lammen, 4-Sep-2023.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))
Theorem	sbbibvv 2368*	Reversal of substitution. (Contributed by AV, 6-Aug-2023.)
⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))
Theorem	cbvsbvf 2369*	Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was part of a former cbvabw 2816 version. (Contributed by GG and WL, 26-Oct-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
Theorem	cleljustALT 2370*	Alternate proof of cleljust 2117. It is kept here and should not be modified because it is referenced on the Metamath Proof Explorer Home Page (mmset.html) as an example of how disjoint variable conditions are inherited by substitutions. (Contributed by NM, 28-Jan-2004.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))
Theorem	cleljustALT2 2371*	Alternate proof of cleljust 2117. Compared with cleljustALT 2370, it uses nfv 1913 followed by equsexv 2269 instead of ax-5 1909 followed by equsexhv 2296, so it uses the idiom Ⅎ𝑥𝜑 instead of 𝜑 → ∀𝑥𝜑 to express nonfreeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))
Theorem	equs5aALT 2372	Alternate proof of equs5a 2465. Uses ax-12 2178 but not ax-13 2380. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs5eALT 2373	Alternate proof of equs5e 2466. Uses ax-12 2178 but not ax-13 2380. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Theorem	axc11r 2374	Same as axc11 2438 but with reversed antecedent. Note the use of ax-12 2178 (and not merely ax12v 2179 as in axc11rv 2266). This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2053 and aecom 2435, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations. As of 2024, it is used in conjunction with ax-13 2380 only, and like that, it should be applied only in niches where indispensable. (Contributed by NM, 25-Jul-2015.)
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	dral1v 2375*	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2447 with a disjoint variable condition, which does not require ax-13 2380. Remark: the corresponding versions for dral2 2446 and drex2 2450 are instances of albidv 1919 and exbidv 1920 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2179. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2141. (Revised by GG, 18-Nov-2024.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	dral1vOLD 2376*	Obsolete version of dral1v 2375 as of 18-Nov-2024. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2179. (Revised by Wolf Lammen, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	drex1v 2377*	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drex1 2449 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 27-Feb-2005.) (Revised by BJ, 17-Jun-2019.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Theorem	drnf1v 2378*	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2451 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2141. (Revised by GG, 18-Nov-2024.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
Theorem	drnf1vOLD 2379*	Obsolete version of drnf1v 2378 as of 18-Nov-2024. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
1.5.4  Axiom scheme ax-13 (Quantified Equality)
Axiom	ax-13 2380	Axiom of Quantified Equality. One of the equality and substitution axioms of predicate calculus with equality. An equivalent way to express this axiom that may be easier to understand is (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧))) (see ax13b 2031). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent ¬ 𝑥 = 𝑦 to hold, 𝑥 and 𝑦 must have different values and thus cannot be the same object-language variable (so they are effectively "distinct variables" even though no $d is present). Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-5 1909, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows. Note that ax-5 1909 cannot prove this because its distinct variable ($d) requirement is not satisfied directly but only indirectly (outside of Metamath) by the argument above. The original version of this axiom was ax-c9 38846 and was replaced with this shorter ax-13 2380 in December 2015. The old axiom is proved from this one as Theorem axc9 2390. The primary purpose of this axiom is to provide a way to introduce the quantifier ∀𝑥 on 𝑦 = 𝑧 even when 𝑥 and 𝑦 are substituted with the same variable. In this case, the first antecedent becomes ¬ 𝑥 = 𝑥 and the axiom still holds. This axiom is mostly used to eliminate conditions requiring set variables be distinct (cf. ax6ev 1969 and ax6e 2391, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations, so direct or indirect application of this axiom is discouraged now. You need to explicitly confirm its use in case you see a sensible application in a niche. After some assisting contributions by others over the years, it was in particular the extensive work of Gino Giotto in 2024 that helped reducing dependencies on this axiom on a large scale. Although this version is shorter, the original version axc9 2390 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2390 is in dvelimh 2458 which converts a distinct variable pair to the distinctor antecedent ¬ ∀𝑥𝑥 = 𝑦. In particular, it is conjectured that it is not possible to prove ax6 2392 from ax6v 1968 without this axiom. This axiom can be weakened if desired by adding distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. To show that, we add these restrictions to Theorem ax13v 2381 and use only ax13v 2381 for further derivations. Thus, ax13v 2381 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2381 (preferred) or ax13 2383 (if the stronger form is needed). This axiom scheme is logically redundant (see ax13w 2136) but is used as an auxiliary axiom scheme to achieve scheme completeness (i.e. so that all possible cases of bundling can be proved; see text linked at mmtheorems.html#ax6dgen 2136). It is not known whether this axiom can be derived from the others. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax13v 2381*	A weaker version of ax-13 2380 with distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. In order to show (with ax13 2383) that this weakening is still adequate, this should be the only theorem referencing ax-13 2380 directly. Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1909. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. Preferably, use the version ax13w 2136 to avoid the propagation of ax-13 2380. (Contributed by NM, 30-Jun-2016.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax13lem1 2382*	A version of ax13v 2381 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2383 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	ax13 2383	Derive ax-13 2380 from ax13v 2381 and Tarski's FOL. This shows that the weakening in ax13v 2381 is still sufficient for a complete system. Preferably, use the weaker ax13w 2136 to avoid the propagation of ax-13 2380. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 2-Jun-2021.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax13lem2 2384*	Lemma for nfeqf2 2385. This lemma is equivalent to ax13v 2381 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
Theorem	nfeqf2 2385*	An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Theorem	dveeq2 2386*	Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	nfeqf1 2387*	An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
Theorem	dveeq1 2388*	Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	nfeqf 2389	A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 38846. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
Theorem	axc9 2390	Derive set.mm's original ax-c9 38846 from the shorter ax-13 2380. Usage is discouraged to avoid uninformed ax-13 2380 propagation. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Theorem	ax6e 2391	At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-4 1807 through ax-9 2118, all axioms other than ax-6 1967 are believed to be theorems of free logic, although the system without ax-6 1967 is not complete in free logic. Usage of this theorem is discouraged because it depends on ax-13 2380. It is preferred to use ax6ev 1969 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2382 became available. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	ax6 2392	Theorem showing that ax-6 1967 follows from the weaker version ax6v 1968. (Even though this theorem depends on ax-6 1967, all references of ax-6 1967 are made via ax6v 1968. An earlier version stated ax6v 1968 as a separate axiom, but having two axioms caused some confusion.) This theorem should be referenced in place of ax-6 1967 so that all proofs can be traced back to ax6v 1968. When possible, use the weaker ax6v 1968 rather than ax6 2392 since the ax6v 1968 derivation is much shorter and requires fewer axioms. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.) Usage of this theorem is discouraged because it depends on ax-13 2380. Use ax6v 1968 instead. (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	axc10 2393	Show that the original axiom ax-c10 38842 can be derived from ax6 2392 and axc7 2321 (on top of propositional calculus, ax-gen 1793, and ax-4 1807). See ax6fromc10 38852 for the rederivation of ax6 2392 from ax-c10 38842. Normally, axc10 2393 should be used rather than ax-c10 38842, except by theorems specifically studying the latter's properties. See bj-axc10v 36759 for a weaker version requiring fewer axioms. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) Usage of this theorem is discouraged because it depends on ax-13 2380. (New usage is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	spimt 2394	Closed theorem form of spim 2395. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Mar-2023.) Usage of this theorem is discouraged because it depends on ax-13 2380. (New usage is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	spim 2395	Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2395 series of theorems requires that only one direction of the substitution hypothesis hold. Usage of this theorem is discouraged because it depends on ax-13 2380. See spimw 1970 for a version requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimed 2396	Deduction version of spime 2397. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) Usage of this theorem is discouraged because it depends on ax-13 2380. Use spimedv 2198 instead. (New usage is discouraged.)
⊢ (𝜒 → Ⅎ𝑥𝜑)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))
Theorem	spime 2397	Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 1971 and spimevw 1994 for weaker versions requiring fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Usage of this theorem is discouraged because it depends on ax-13 2380. Use spimefv 2199 instead. (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	spimv 2398*	A version of spim 2395 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2240 and spimvw 1995 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2380. Use spimvw 1995 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimvALT 2399*	Alternate proof of spimv 2398. Note that it requires only ax-1 6 through ax-5 1909 together with ax6e 2391. Currently, proofs derive from ax6v 1968, but if ax-6 1967 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2141. (Revised by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimev 2400*	Distinct-variable version of spime 2397. (Contributed by NM, 10-Jan-1993.) Usage of this theorem is discouraged because it depends on ax-13 2380. Use spimevw 1994 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)