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Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremr19.29d2r 2601 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)       (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))

Theoremr19.29vva 2602* A commonly used pattern based on r19.29 2594, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜒)

Theoremr19.32r 2603 One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
𝑥𝜑       ((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))

Theoremr19.30dc 2604 Restricted quantifier version of 19.30dc 1607. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
((∀𝑥𝐴 (𝜑𝜓) ∧ DECID𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Theoremr19.32vr 2605* One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence, as seen at r19.32vdc 2606. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))

Theoremr19.32vdc 2606* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
(DECID 𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓)))

Theoremr19.35-1 2607 Restricted quantifier version of 19.35-1 1604. (Contributed by Jim Kingdon, 4-Jun-2018.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Theoremr19.36av 2608* One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if 𝐴 has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.)
(∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))

Theoremr19.37 2609 Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if 𝐴 has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜑       (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))

Theoremr19.37av 2610* Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))

Theoremr19.40 2611 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))

Theoremr19.41 2612 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
𝑥𝜓       (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Theoremr19.41v 2613* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Theoremr19.42v 2614* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝐴 𝜓))

Theoremr19.43 2615 Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
(∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Theoremr19.44av 2616* One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when 𝐴 is empty. (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑𝜓))

Theoremr19.45av 2617* Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)
(∃𝑥𝐴 (𝜑𝜓) → (𝜑 ∨ ∃𝑥𝐴 𝜓))

Theoremralcomf 2618* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)

Theoremrexcomf 2619* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴    &   𝑥𝐵       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)

Theoremralcom 2620* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥𝐴 𝜑)

Theoremrexcom 2621* Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)

Theoremrexcom13 2622* Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)

Theoremrexrot4 2623* Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)

Theoremralcom3 2624 A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))

Theoremreean 2625* Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))

Theoremreeanv 2626* Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))

Theorem3reeanv 2627* Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))

Theoremnfreu1 2628 𝑥 is not free in ∃!𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
𝑥∃!𝑥𝐴 𝜑

Theoremnfrmo1 2629 𝑥 is not free in ∃*𝑥𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
𝑥∃*𝑥𝐴 𝜑

Theoremnfreudxy 2630* Not-free deduction for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)

Theoremnfreuxy 2631* Not-free for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
𝑥𝐴    &   𝑥𝜑       𝑥∃!𝑦𝐴 𝜑

Theoremrabid 2632 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 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))

Theoremrabid2 2633* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)

Theoremrabbi 2634 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2700. (Contributed by NM, 25-Nov-2013.)
(∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabswap 2635 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
{𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}

Theoremnfrab1 2636 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
𝑥{𝑥𝐴𝜑}

Theoremnfrabxy 2637* A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
𝑥𝜑    &   𝑥𝐴       𝑥{𝑦𝐴𝜑}

Theoremreubida 2638 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))

Theoremreubidva 2639* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))

Theoremreubidv 2640* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))

Theoremreubiia 2641 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)
(𝑥𝐴 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)

Theoremreubii 2642 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 22-Oct-1999.)
(𝜑𝜓)       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)

Theoremrmobida 2643 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))

Theoremrmobidva 2644* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))

Theoremrmobidv 2645* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))

Theoremrmobiia 2646 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
(𝑥𝐴 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)

Theoremrmobii 2647 Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 16-Jun-2017.)
(𝜑𝜓)       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)

Theoremraleqf 2648 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))

Theoremrexeqf 2649 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Theoremreueq1f 2650 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))

Theoremrmoeq1f 2651 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))

Theoremraleq 2652* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))

Theoremrexeq 2653* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
(𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Theoremreueq1 2654* Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))

Theoremrmoeq1 2655* Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))

Theoremraleqi 2656* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑)

Theoremrexeqi 2657* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐴 = 𝐵       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)

Theoremraleqdv 2658* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))

Theoremrexeqdv 2659* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))

Theoremraleqbi1dv 2660* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))

Theoremrexeqbi1dv 2661* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))

Theoremreueqd 2662* Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))

Theoremrmoeqd 2663* Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜓))

Theoremraleqbidv 2664* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremrexeqbidv 2665* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremraleqbidva 2666* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremrexeqbidva 2667* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremmormo 2668 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝜑 → ∃*𝑥𝐴 𝜑)

Theoremreu5 2669 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
(∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐴 𝜑))

Theoremreurex 2670 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
(∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)

Theoremreurmo 2671 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
(∃!𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)

Theoremrmo5 2672 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))

Theoremnrexrmo 2673 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
(¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)

Theoremcbvralf 2674 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrexf 2675 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvral 2676* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrex 2677* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvreu 2678* Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)

Theoremcbvrmo 2679* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)

Theoremcbvralv 2680* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrexv 2681* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvreuv 2682* Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)

Theoremcbvrmov 2683* Change the bound variable of a restricted at-most-one quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)

Theoremcbvralvw 2684* Version of cbvralv 2680 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Theoremcbvrexvw 2685* Version of cbvrexv 2681 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Theoremcbvreuvw 2686* Version of cbvreuv 2682 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)

Theoremcbvraldva2 2687* 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.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒))

Theoremcbvrexdva2 2688* 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.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))

Theoremcbvraldva 2689* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐴 𝜒))

Theoremcbvrexdva 2690* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))

Theoremcbvral2v 2691* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvrex2v 2692* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvral3v 2693* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
(𝑥 = 𝑤 → (𝜑𝜒))    &   (𝑦 = 𝑣 → (𝜒𝜃))    &   (𝑧 = 𝑢 → (𝜃𝜓))       (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)

Theoremcbvralsv 2694* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)

Theoremcbvrexsv 2695* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)

Theoremsbralie 2696* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦𝑥 𝜓)

Theoremrabbiia 2697 Equivalent wff's yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}

Theoremrabbii 2698 Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 2701. (Contributed by Peter Mazsa, 1-Nov-2019.)
(𝜑𝜓)       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}

Theoremrabbidva2 2699* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabbidva 2700* Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

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