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	equsalv 2301*	An equivalence related to implicit substitution. Version of equsal 2447 with a disjoint variable condition, which does not require ax-13 2402. See equsalvw 2023 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv 2302. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	equsexv 2302*	An equivalence related to implicit substitution. Version of equsex 2448 with a disjoint variable condition, which does not require ax-13 2402. See equsexvw 2024 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2301. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2174. (Revised by GG, 18-Nov-2024.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	sbft 2303	Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbf 2304	Substitution for a variable not free in a wff does not affect it. For a version requiring disjoint variables but fewer axioms, see sbv 2120. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
Theorem	sbf2 2305	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
⊢ ([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	sbh 2306	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
Theorem	hbs1 2307*	The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 26-May-1993.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	nfs1f 2308	If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
Theorem	sb5 2309*	Alternate definition of substitution when variables are disjoint. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1v 2119 and even needs no disjoint variable condition, see sb1 2508. Theorem sb5f 2528 replaces the disjoint variable condition with a nonfreeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Revised by Wolf Lammen, 4-Sep-2023.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
Theorem	equs5av 2310*	A property related to substitution that replaces the distinctor from equs5 2490 to a disjoint variable condition. Version of equs5a 2487 with a disjoint variable condition, which does not require ax-13 2402. See also sbalex 2276. (Contributed by NM, 2-Feb-2007.) (Revised by GG, 15-Dec-2023.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	2sb5 2311*	Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
Theorem	dfsb7 2312*	An alternate definition of proper substitution df-sb 2090. By introducing a dummy variable 𝑦 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑡, 𝑥, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑦 effectively insulates 𝑡 from 𝑥. To achieve this, we use a chain of two substitutions in the form of sb5 2309, first 𝑦 for 𝑥 then 𝑡 for 𝑦. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2740. Theorem sb7h 2556 provides a version where 𝜑 and 𝑦 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2090. (Revised by BJ, 25-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
Theorem	sbn 2313	Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) Revise df-sb 2090. (Revised by BJ, 25-Dec-2020.)
⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑)
Theorem	sbex 2314*	Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Theorem	nf5 2315	Alternate definition of df-nf 1803. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1803 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Theorem	nf6 2316	An alternate definition of df-nf 1803. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
Theorem	nf5d 2317	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nf5di 2318	Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either Ⅎ𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)
⊢ (𝜑 → Ⅎ𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	19.9h 2319	A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	19.21h 2320	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See also 19.21 2241 and 19.21v 1958. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	19.23h 2321	Theorem 19.23 of [Margaris] p. 90. See 19.23 2245. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	exlimih 2322	Inference associated with 19.23 2245. See exlimiv 1949 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimdh 2323	Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜒 → ∀𝑥𝜒)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	equsalhw 2324*	Version of equsalh 2450 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	equsexhv 2325*	An equivalence related to implicit substitution. Version of equsexh 2451 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	hba1 2326	The setvar 𝑥 is not free in ∀𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Oct-2021.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	hbnt 2327	Closed theorem version of bound-variable hypothesis builder hbn 2328. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 14-Oct-2021.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
Theorem	hbn 2328	If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
Theorem	hbnd 2329	Deduction form of bound-variable hypothesis builder hbn 2328. (Contributed by NM, 3-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
Theorem	hbim1 2330	A closed form of hbim 2332. (Contributed by NM, 2-Jun-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	hbimd 2331	Deduction form of bound-variable hypothesis builder hbim 2332. (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 2332	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 2333	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 2334	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 2335	Introduction of implication into substitution. (Contributed by NM, 14-May-1993.)
⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
Theorem	sbim 2336	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 2337	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 2174. (Revised by GG, 20-Nov-2024.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sblim 2338	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 2339	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 2340	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 2341	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 2342	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 2343	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 2344*	Move nonfree predicate in and out of substitution; see sbal 2202 and sbex 2314. (Contributed by BJ, 2-May-2019.) (Proof shortened by Wolf Lammen, 2-May-2025.)
⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
Theorem	sbiev 2345*	Conversion of implicit substitution to explicit substitution. Version of sbie 2532 with a disjoint variable condition, not requiring ax-13 2402. See sbievw 2126 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 2174 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 2346*	Obsolete version of sbiev 2345 as of 24-Aug-2025. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2174 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 2347*	Conversion of implicit substitution to explicit substitution (deduction version of sbiev 2345). Version of sbied 2533 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) Avoid ax-13 2402. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	axc7 2348	Show that the original axiom ax-c7 39473 can be derived from ax-10 2174 (hbn1 2175), sp 2217 and propositional calculus. See ax10fromc7 39483 for the rederivation of ax-10 2174 from ax-c7 39473. Normally, axc7 2348 should be used rather than ax-c7 39473, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
Theorem	axc7e 2349	Abbreviated version of axc7 2348 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 2350	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 2351	A closed version of 19.9h 2319. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
Theorem	axc4 2352	Show that the original axiom ax-c4 39472 can be derived from ax-4 1828 (alim 1829), ax-10 2174 (hbn1 2175), sp 2217 and propositional calculus. See ax4fromc4 39482 for the rederivation of ax-4 1828 from ax-c4 39472. 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 2353	Inference version of axc4 2352. (Contributed by NM, 3-Jan-1993.)
⊢ (∀𝑥𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	nfal 2354	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 2355	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 2355, hbex 2356. (Revised by Wolf Lammen, 16-Oct-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃𝑦𝜑
Theorem	hbex 2356	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 2355, hbex 2356. (Revised by Wolf Lammen, 16-Oct-2021.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
Theorem	nfnf 2357	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 2358	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 2377 and r19.12sn 4678. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)
Theorem	nfald 2359	Deduction form of nfal 2354. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 16-Oct-2021.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	nfexd 2360	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 2361*	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 2553 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2402. (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	sbco2v 2362*	A composition law for substitution. Version of sbco2 2541 with disjoint variable conditions but not requiring ax-13 2402. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 29-Apr-2023.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	aaan 2363	Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2174. (Revised by GG, 21-Nov-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Theorem	eeor 2364	Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2174. (Revised by GG, 21-Nov-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
Theorem	cbv3v 2365*	Rule used to change bound variables, using implicit substitution. Version of cbv3 2427 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	cbv1v 2366*	Rule used to change bound variables, using implicit substitution. Version of cbv1 2432 with a disjoint variable condition, which does not require ax-13 2402. (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 2367*	Rule used to change bound variables, using implicit substitution. Version of cbv2 2433 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2402. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvaldw 2368*	Deduction used to change bound variables, using implicit substitution. Version of cbvald 2437 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2402. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvexdw 2369*	Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2438 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2402. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	cbv3hv 2370*	Rule used to change bound variables, using implicit substitution. Version of cbv3h 2434 with a disjoint variable condition on 𝑥, 𝑦, which does not require ax-13 2402. Was used in a proof of axc11n 2456 (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 2371*	Rule used to change bound variables, using implicit substitution. Version of cbval 2428 with a disjoint variable condition, which does not require ax-13 2402. See cbvalvw 2055 for a version with two more disjoint variable conditions, requiring fewer axioms, and cbvalv 2430 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 2372*	Rule used to change bound variables, using implicit substitution. Version of cbvex 2429 with a disjoint variable condition, which does not require ax-13 2402. See cbvexvw 2056 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2431 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 2373*	Rule used to change bound variables, using implicit substitution. Version of cbval2 2441 with a disjoint variable condition, which does not require ax-13 2402. (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 2374*	Rule used to change bound variables, using implicit substitution. Version of cbvex2 2442 with a disjoint variable condition, which does not require ax-13 2402. (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 2375*	Proof of dvelimh 2480 without using ax-13 2402 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 2376*	Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1963 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Theorem	19.12vv 2377*	Special case of 19.12 2358 where its converse holds. See 19.12vvv 2013 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 2378	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 2379*	Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1968 and 19.42v 1972. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1974. (Contributed by NM, 26-Jul-1995.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Theorem	eeeanv 2380*	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 2381*	Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv 1975. (Contributed by NM, 31-Jul-1995.) Remove disjoint variable conditions on 𝑦, 𝑧 and 𝑥, 𝑤. (Revised by Eric Schmidt, 26-Oct-2025.)
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
Theorem	ee4anvOLD 2382*	Obsolete version of ee4anv 2381 as of 26-Oct-2025. (Contributed by NM, 31-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
Theorem	sb8v 2383*	Substitution of variable in universal quantifier. Version of sb8f 2384 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2211. (Contributed by SN, 5-Dec-2024.)
⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb8f 2384*	Substitution of variable in universal quantifier. Version of sb8 2547 with a disjoint variable condition, not requiring ax-10 2174 or ax-13 2402. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2174. (Revised by SN, 5-Dec-2024.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb8ef 2385*	Substitution of variable in existential quantifier. Version of sb8e 2548 with a disjoint variable condition, not requiring ax-13 2402. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Theorem	2sb8ef 2386*	An equivalent expression for double existence. Version of 2sb8e 2560 with more disjoint variable conditions, not requiring ax-13 2402. (Contributed by Wolf Lammen, 28-Jan-2023.)
⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Theorem	sb6rfv 2387*	Reversed substitution. Version of sb6rf 2498 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 2388*	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 2402. (Revised by Wolf Lammen, 30-Jan-2023.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
Theorem	exsb 2389*	An equivalent expression for existence. One direction (exsbim 2021) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2402. (Revised by Wolf Lammen, 16-Oct-2022.)
⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	2exsb 2390*	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 2391*	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 2392*	Reversal of substitution. (Contributed by AV, 6-Aug-2023.)
⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))
Theorem	cbvsbvf 2393*	Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was part of a former cbvabw 2832 version. (Contributed by GG and WL, 26-Oct-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
Theorem	cleljustALT 2394*	Alternate proof of cleljust 2150. 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 2395*	Alternate proof of cleljust 2150. Compared with cleljustALT 2394, it uses nfv 1933 followed by equsexv 2302 instead of ax-5 1929 followed by equsexhv 2325, 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 2396	Alternate proof of equs5a 2487. Uses ax-12 2211 but not ax-13 2402. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs5eALT 2397	Alternate proof of equs5e 2488. Uses ax-12 2211 but not ax-13 2402. (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 2398	Same as axc11 2460 but with reversed antecedent. Note the use of ax-12 2211 (and not merely ax12v 2212 as in axc11rv 2299). This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2074 and aecom 2457, 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 2402 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 2399*	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2469 with a disjoint variable condition, which does not require ax-13 2402. Remark: the corresponding versions for dral2 2468 and drex2 2472 are instances of albidv 1939 and exbidv 1940 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2212. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2174. (Revised by GG, 18-Nov-2024.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	drex1v 2400*	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drex1 2471 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 27-Feb-2005.) (Revised by BJ, 17-Jun-2019.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))