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	r19.41 3301	Restricted quantifier version of 19.41 2235. See r19.41v 3300 for a version with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 1-Nov-2010.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))
Theorem	r19.41vv 3302*	Version of r19.41v 3300 with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ 𝜓))
Theorem	r19.42v 3303*	Restricted quantifier version of 19.42v 1954 (see also 19.42 2236). (Contributed by NM, 27-May-1998.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓))
Theorem	r19.43 3304	Restricted quantifier version of 19.43 1883. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
Theorem	r19.44v 3305*	One direction of a restricted quantifier version of 19.44 2237. The other direction holds when 𝐴 is nonempty, see r19.44zv 4407. (Contributed by NM, 2-Apr-2004.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))
Theorem	r19.45v 3306*	Restricted quantifier version of one direction of 19.45 2238. The other direction holds when 𝐴 is nonempty, see r19.45zv 4406. (Contributed by NM, 2-Apr-2004.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
Theorem	ralcom 3307*	Commutation of restricted universal quantifiers. See ralcom2 3316 for a version without disjoint variable condition on 𝑥, 𝑦. This theorem should be used in place of ralcom2 3316 since it depends on a smaller set of axioms. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexcom 3308*	Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof shortened by BJ, 26-Aug-2023.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
Theorem	rexcomOLD 3309*	Obsolete version of rexcom 3308 as of 26-Aug-2023. Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
Theorem	ralcomf 3310*	Commutation of restricted universal quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexcomf 3311*	Commutation of restricted existential quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
Theorem	ralcom13 3312*	Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexcom13 3313*	Swap first and third restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
Theorem	ralrot3 3314*	Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
Theorem	rexrot4 3315*	Rotate four restricted existential quantifiers twice. (Contributed by NM, 8-Apr-2015.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	ralcom2 3316*	Commutation of restricted universal quantifiers. Note that 𝑥 and 𝑦 need not be disjoint (this makes the proof longer). This theorem relies on the full set of axioms up to ax-ext 2770 and it should no longer be used. Usage of ralcom 3307 is highly encouraged. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (New usage is discouraged.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ralcom3 3317	A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))
Theorem	reeanlem 3318*	Lemma factoring out common proof steps of reeanv 3320 and reean 3319. (Contributed by Wolf Lammen, 20-Aug-2023.)
⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜓)))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))
Theorem	reean 3319*	Rearrange restricted existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))
Theorem	reeanv 3320*	Rearrange restricted existential quantifiers. (Contributed by NM, 9-May-1999.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))
Theorem	3reeanv 3321*	Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒))
Theorem	2ralor 3322*	Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))
Theorem	nfreu1 3323	The setvar 𝑥 is not free in ∃!𝑥 ∈ 𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜑
Theorem	nfrmo1 3324	The setvar 𝑥 is not free in ∃*𝑥 ∈ 𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝜑
Theorem	nfreud 3325	Deduction version of nfreu 3329. Usage of this theorem is discouraged because it depends on ax-13 2379. (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 3326	Deduction version of nfrmo 3330. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓)
Theorem	nfreuw 3327*	Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3329 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
Theorem	nfrmow 3328*	Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3330 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	nfreu 3329	Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfreuw 3327 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 3330	Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfrmow 3328 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	rabid 3331	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 3332	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 3333	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
Theorem	rabid2 3334*	An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rabid2f 3335	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	rabbi 3336	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 3425. (Contributed by NM, 25-Nov-2013.)
⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	nfrab1 3337	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	nfrabw 3338*	A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3339 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	nfrab 3339	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 2379. Use the weaker nfrabw 3338 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 3340	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 3341*	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 3342*	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 3343	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 3344	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 3345	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 3346*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	rmobidv 3347*	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 3348	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 3349	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 3350	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 3351	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 3352	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 3353	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 3354*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2142, ax-11 2158, and ax-12 2175 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 3355*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) Remove usage of ax-10 2142, ax-11 2158, and ax-12 2175 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	raleqbi1dv 3356*	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 3357*	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 3358*	Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2142, ax-11 2158, and ax-12 2175. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	rexeq 3359*	Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) Remove usage of ax-10 2142, ax-11 2158, and ax-12 2175. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
Theorem	reueq1 3360*	Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2142, ax-11 2158, and ax-12 2175. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
Theorem	rmoeq1 3361*	Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2142, ax-11 2158, and ax-12 2175. (Revised by Steven Nguyen, 30-Apr-2023.)
⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	raleqi 3362*	Equality inference for restricted universal quantifier. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)
Theorem	rexeqi 3363*	Equality inference for restricted existential quantifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)
Theorem	raleqdv 3364*	Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	rexeqdv 3365*	Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
Theorem	reueqd 3366*	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 3367*	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 3368	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 3369	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 3370*	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 3371*	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 3372*	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 3373*	Alternate proof of raleleq 3372 using ralel 3117, 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	mormo 3374	Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	reu5 3375	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 3376	Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	2reu2rex 3377	Double restricted existential uniqueness, analogous to 2eu2ex 2705. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	reurmo 3378	Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	rmo5 3379	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 3380	Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	reueubd 3381*	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 3382*	Rule used to change bound variables, using implicit substitution. Version of cbvralf 3385 with a disjoint variable condition, which does not require ax-10 2142, ax-13 2379. (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 3383*	Obsolete version of cbvralfw 3382 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 3384*	Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3386 with a disjoint variable condition, which does not require ax-13 2379. (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 3385	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvralfw 3382 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 3386	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvrexfw 3384 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 3387*	Rule used to change bound variables, using implicit substitution. Version of cbvral 3392 with a disjoint variable condition, which does not require ax-13 2379. (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 3388*	Rule used to change bound variables, using implicit substitution. Version of cbvrex 3393 with a disjoint variable condition, which does not require ax-13 2379. (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	cbvreuw 3389*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3394 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmow 3390*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3395 with a disjoint variable condition, which does not require ax-10 2142, ax-13 2379. (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	cbvrmowOLD 3391*	Obsolete version of cbvrmow 3390 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 3392*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvralw 3387 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 3393*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvrexw 3388 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 3394*	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 2379. Use the weaker cbvreuw 3389 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 3395*	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 2379. Use the weaker cbvrmow 3390 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 3396*	Change the bound variable of a restricted universal quantifier using implicit substitution. Version of cbvralv 3399 with a disjoint variable condition, which does not require ax-10 2142, ax-11 2158, ax-12 2175, ax-13 2379. (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 3397*	Change the bound variable of a restricted existential quantifier using implicit substitution. Version of cbvrexv 3400 with a disjoint variable condition, which does not require ax-10 2142, ax-11 2158, ax-12 2175, ax-13 2379. (Contributed by NM, 2-Jun-1998.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuvw 3398*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3401 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvralv 3399*	Change the bound variable of a restricted universal quantifier using implicit substitution. See cbvralvw 3396 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvralvw 3396 when possible. (Contributed by NM, 28-Jan-1997.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrexv 3400*	Change the bound variable of a restricted existential quantifier using implicit substitution. See cbvrexvw 3397 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvrexvw 3397 when possible. (Contributed by NM, 2-Jun-1998.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)