P. 34 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3301-3400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfrmo1 3301	The setvar 𝑥 is not free in ∃*𝑥 ∈ 𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝜑
Theorem	nfreud 3302	Deduction version of nfreu 3308. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓)
Theorem	nfrmod 3303	Deduction version of nfrmo 3309. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓)
Theorem	nfrmow 3304*	Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3309 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2709. (Revised by Wolf Lammen, 21-Nov-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	nfreuw 3305*	Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3308 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2709. (Revised by Wolf Lammen, 21-Nov-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
Theorem	nfreuwOLD 3306*	Obsolete version of nfreuw 3305 as of 21-Nov-2024. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
Theorem	nfrmowOLD 3307*	Obsolete version of nfrmow 3304 as of 21-Nov-2024. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	nfreu 3308	Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfreuw 3305 when possible. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
Theorem	nfrmo 3309	Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfrmow 3304 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	rabid 3310	An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	rabrab 3311	Abstract builder restricted to another restricted abstract builder. (Contributed by Thierry Arnoux, 30-Aug-2017.)
⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}
Theorem	rabidim1 3312	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
Theorem	rabid2f 3313	An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rabid2 3314*	An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2024.)
⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rabid2OLD 3315*	Obsolete version of rabid2 3314 as of 24-11-2024. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rabbi 3316	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 3413. (Contributed by NM, 25-Nov-2013.)
⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	nfrab1 3317	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	nfrabw 3318*	A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3320 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	nfrabwOLD 3319*	Obsolete version of nfrabw 3318 as of 23-Nov2024. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	nfrab 3320	A variable not free in a wff remains so in a restricted class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfrabw 3318 when possible. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	reubida 3321	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	reubidva 3322*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	reubidv 3323*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	reubiia 3324	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓)
Theorem	reubii 3325	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 22-Oct-1999.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓)
Theorem	rmobida 3326	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	rmobidva 3327*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) Avoid ax-6 1971, ax-7 2011, ax-12 2171. (Revised by Wolf Lammen, 23-Nov-2024.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	rmobidvaOLD 3328*	Obsolete version of rmobidv 3329 as of 23-Nov-2024. (Contributed by NM, 16-Jun-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	rmobidv 3329*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	rmobiia 3330	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)
Theorem	rmobii 3331	Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)
Theorem	raleqf 3332	Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	rexeqf 3333	Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
Theorem	reueq1f 3334	Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
Theorem	rmoeq1f 3335	Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	raleqbidv 3336*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexeqbidv 3337*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	raleqbidvv 3338*	Version of raleqbidv 3336 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2816. (Contributed by BJ, 22-Sep-2024.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexeqbidvv 3339*	Version of rexeqbidv 3337 with additional disjoint variable conditions, not requiring ax-8 2108 nor df-clel 2816. (Contributed by Wolf Lammen, 25-Sep-2024.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	raleqbi1dv 3340*	Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) (Proof shortened by Steven Nguyen, 5-May-2023.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	rexeqbi1dv 3341*	Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) (Proof shortened by Steven Nguyen, 5-May-2023.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓))
Theorem	raleq 3342*	Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	rexeq 3343*	Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
Theorem	reueq1 3344*	Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
Theorem	rmoeq1 3345*	Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2137, ax-11 2154, and ax-12 2171. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	raleqi 3346*	Equality inference for restricted universal quantifier. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)
Theorem	rexeqi 3347*	Equality inference for restricted existential quantifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)
Theorem	raleqdv 3348*	Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	rexeqdv 3349*	Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
Theorem	reueqd 3350*	Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
Theorem	rmoeqd 3351*	Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓))
Theorem	raleqbid 3352	Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexeqbid 3353	Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	raleqbidva 3354*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexeqbidva 3355*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	raleleq 3356*	All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.)
⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	raleleqALT 3357*	Alternate proof of raleleq 3356 using ralel 3075, being longer and using more axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	moel 3358*	"At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2154. (Revised by Wolf Lammen, 23-Nov-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)
Theorem	moelOLD 3359*	Obsolete version of moel 3358 as of 23-Nov-2024. (Contributed by Thierry Arnoux, 26-Jul-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)
Theorem	mormo 3360	Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	reu5 3361	Restricted uniqueness in terms of "at most one". (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
Theorem	reurex 3362	Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	2reu2rex 3363	Double restricted existential uniqueness, analogous to 2eu2ex 2645. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	reurmo 3364	Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	rmo5 3365	Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
Theorem	nrexrmo 3366	Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	reueubd 3367*	Restricted existential uniqueness is equivalent to existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.)
⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓))
Theorem	cbvralfw 3368*	Rule used to change bound variables, using implicit substitution. Version of cbvralf 3371 with a disjoint variable condition, which does not require ax-10 2137, ax-13 2372. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 23-May-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvralfwOLD 3369*	Obsolete version of cbvralfw 3368 as of 23-May-2024. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrexfw 3370*	Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3372 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvralf 3371	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvralfw 3368 when possible. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrexf 3372	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvrexfw 3370 when possible. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvralw 3373*	Rule used to change bound variables, using implicit substitution. Version of cbvral 3379 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrexw 3374*	Rule used to change bound variables, using implicit substitution. Version of cbvrex 3380 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmow 3375*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3382 with a disjoint variable condition, which does not require ax-10 2137, ax-13 2372. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 23-May-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuw 3376*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3381 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2137. (Revised by Wolf Lammen, 10-Dec-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuwOLD 3377*	Obsolete version of cbvreuw 3376 as of 10-Dec-2024. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmowOLD 3378*	Obsolete version of cbvrmow 3375 as of 23-May-2024. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvral 3379*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvralw 3373 when possible. (Contributed by NM, 31-Jul-2003.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrex 3380*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvrexw 3374 when possible. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreu 3381*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvreuw 3376 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmo 3382*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvrmow 3375, cbvrmovw 3385 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvralvw 3383*	Change the bound variable of a restricted universal quantifier using implicit substitution. Version of cbvralv 3388 with a disjoint variable condition, which does not require ax-10 2137, ax-11 2154, ax-12 2171, ax-13 2372. (Contributed by NM, 28-Jan-1997.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrexvw 3384*	Change the bound variable of a restricted existential quantifier using implicit substitution. Version of cbvrexv 3389 with a disjoint variable condition, which does not require ax-10 2137, ax-11 2154, ax-12 2171, ax-13 2372. (Contributed by NM, 2-Jun-1998.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmovw 3385*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov 3391 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuvw 3386*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3390 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuvwOLD 3387*	Obsolete version of cbvreuvw 3386 as of 30-Sep-2024. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvralv 3388*	Change the bound variable of a restricted universal quantifier using implicit substitution. See cbvralvw 3383 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvralvw 3383 when possible. (Contributed by NM, 28-Jan-1997.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrexv 3389*	Change the bound variable of a restricted existential quantifier using implicit substitution. See cbvrexvw 3384 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvrexvw 3384 when possible. (Contributed by NM, 2-Jun-1998.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuv 3390*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3386 for a version without ax-13 2372, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvreuvw 3386 when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmov 3391*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvraldva2 3392*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒))
Theorem	cbvrexdva2 3393*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2023.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
Theorem	cbvraldva 3394*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))
Theorem	cbvrexdva 3395*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒))
Theorem	cbvral2vw 3396*	Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3399 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
Theorem	cbvrex2vw 3397*	Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3400 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
Theorem	cbvral3vw 3398*	Change bound variables of triple restricted universal quantification, using implicit substitution. Version of cbvral3v 3401 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 10-May-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
Theorem	cbvral2v 3399*	Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvral2vw 3396 when possible. (Contributed by NM, 10-Aug-2004.) (New usage is discouraged.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
Theorem	cbvrex2v 3400*	Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvrex2vw 3397 when possible. (Contributed by FL, 2-Jul-2012.) (New usage is discouraged.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)