P. 29 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	vtocleg 2801*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜑)
Theorem	vtoclegft 2802*	Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2803.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)
Theorem	vtoclef 2803*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	vtocle 2804*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	vtoclri 2805*	Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ ∀𝑥 ∈ 𝐵 𝜑    ⇒   ⊢ (𝐴 ∈ 𝐵 → 𝜓)
Theorem	spcimgft 2806	A closed version of spcimgf 2810. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))
Theorem	spcgft 2807	A closed version of spcgf 2812. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))
Theorem	spcimegft 2808	A closed version of spcimegf 2811. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑)))
Theorem	spcegft 2809	A closed version of spcegf 2813. (Contributed by Jim Kingdon, 22-Jun-2018.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑)))
Theorem	spcimgf 2810	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))
Theorem	spcimegf 2811	Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))
Theorem	spcgf 2812	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))
Theorem	spcegf 2813	Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))
Theorem	spcimdv 2814*	Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	spcdv 2815*	Rule of specialization, using implicit substitution. Analogous to rspcdv 2837. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	spcimedv 2816*	Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))
Theorem	spcgv 2817*	Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))
Theorem	spcegv 2818*	Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))
Theorem	spcedv 2819*	Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ V)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	spc2egv 2820*	Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝜓 → ∃𝑥∃𝑦𝜑))
Theorem	spc2gv 2821*	Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓))
Theorem	spc3egv 2822*	Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑))
Theorem	spc3gv 2823*	Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓))
Theorem	spcv 2824*	Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spcev 2825*	Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜓 → ∃𝑥𝜑)
Theorem	spc2ev 2826*	Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜓 → ∃𝑥∃𝑦𝜑)
Theorem	rspct 2827*	A closed version of rspc 2828. (Contributed by Andrew Salmon, 6-Jun-2011.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))
Theorem	rspc 2828*	Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
Theorem	rspce 2829*	Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	rspcv 2830*	Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
Theorem	rspccv 2831*	Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))
Theorem	rspcva 2832*	Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝜑) → 𝜓)
Theorem	rspccva 2833*	Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((∀𝑥 ∈ 𝐵 𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓)
Theorem	rspcev 2834*	Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	rspcimdv 2835*	Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
Theorem	rspcimedv 2836*	Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
Theorem	rspcdv 2837*	Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
Theorem	rspcedv 2838*	Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))
Theorem	rspcdva 2839*	Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	rspcedvd 2840*	Restricted existential specialization, using implicit substitution. Variant of rspcedv 2838. (Contributed by AV, 27-Nov-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	rspcime 2841*	Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	rspceaimv 2842*	Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))
Theorem	rspcedeq1vd 2843*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2840 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)
Theorem	rspcedeq2vd 2844*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2840 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)
Theorem	rspc2 2845*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓))
Theorem	rspc2gv 2846*	Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓))
Theorem	rspc2v 2847*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓))
Theorem	rspc2va 2848*	2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑) → 𝜓)
Theorem	rspc2ev 2849*	2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝜓) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 𝜑)
Theorem	rspc3v 2850*	3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 𝜑 → 𝜓))
Theorem	rspc3ev 2851*	3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) ∧ 𝜓) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑇 𝜑)
Theorem	rspceeqv 2852*	Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶)
Theorem	eqvinc 2853*	A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
Theorem	eqvincg 2854*	A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
Theorem	eqvincf 2855	A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
Theorem	alexeq 2856*	Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. (Contributed by NM, 2-Mar-1995.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	ceqex 2857*	Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
Theorem	ceqsexg 2858*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))
Theorem	ceqsexgv 2859*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))
Theorem	ceqsrexv 2860*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))
Theorem	ceqsrexbv 2861*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	ceqsrex2v 2862*	Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))
Theorem	clel2 2863*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
Theorem	clel3g 2864*	An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))
Theorem	clel3 2865*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))
Theorem	clel4 2866*	An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))
Theorem	clel5 2867*	Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.) (Revised by Steven Nguyen, 19-May-2023.)
⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
Theorem	pm13.183 2868*	Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only 𝐴 is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
Theorem	rr19.3v 2869*	Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rr19.28v 2870*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	elabgt 2871*	Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2876.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elabgf 2872	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elabf 2873*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elab 2874*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elabd 2875*	Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝑋. (Contributed by RP, 12-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ V)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	elabg 2876*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab2g 2877*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜓))
Theorem	elab2 2878*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓)
Theorem	elab4g 2879*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ V ∧ 𝜓))
Theorem	elab3gf 2880	Membership in a class abstraction, with a weaker antecedent than elabgf 2872. (Contributed by NM, 6-Sep-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab3g 2881*	Membership in a class abstraction, with a weaker antecedent than elabg 2876. (Contributed by NM, 29-Aug-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab3 2882*	Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
⊢ (𝜓 → 𝐴 ∈ V)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elrabi 2883*	Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)
Theorem	elrabf 2884	Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	elrab3t 2885*	Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2887.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))
Theorem	elrab 2886*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	elrab3 2887*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))
Theorem	elrabd 2888*	Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2886. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	elrab2 2889*	Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑}    ⇒   ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	ralab 2890*	Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))
Theorem	ralrab 2891*	Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
Theorem	rexab 2892*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))
Theorem	rexrab 2893*	Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
Theorem	ralab2 2894*	Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒))
Theorem	ralrab2 2895*	Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))
Theorem	rexab2 2896*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒))
Theorem	rexrab2 2897*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒))
Theorem	abidnf 2898*	Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
Theorem	dedhb 2899*	A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 1534 and nfab 2317 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2898 is useful. (Contributed by NM, 8-Dec-2006.)
⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ (Ⅎ𝑥𝐴 → 𝜑)
Theorem	eqeu 2900*	A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓 ∧ ∀𝑥(𝜑 → 𝑥 = 𝐴)) → ∃!𝑥𝜑)