P. 37 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3601-3700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rspc 3601*	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 3602*	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	rspcimdv 3603*	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 3604*	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 3605*	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 3606*	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	rspcebdv 3607*	Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒))
Theorem	rspcdv2 3608*	Restricted specialization, using implicit substitution. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	rspcv 3609*	Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2138, ax-11 2155, ax-12 2172. (Revised by SN, 12-Dec-2023.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
Theorem	rspccv 3610*	Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))
Theorem	rspcva 3611*	Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝜑) → 𝜓)
Theorem	rspccva 3612*	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 3613*	Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2138, ax-11 2155, ax-12 2172. (Revised by SN, 12-Dec-2023.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	rspcdva 3614*	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 3615*	Restricted existential specialization, using implicit substitution. Variant of rspcedv 3606. (Contributed by AV, 27-Nov-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	rspcedvdw 3616*	Version of rspcedvd 3615 where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on 𝜑, 𝑥. (Contributed by SN, 20-Aug-2024.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	rspcime 3617*	Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	rspceaimv 3618*	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 3619*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3615 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 3620*	Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3615 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 3621*	Restricted specialization with two quantifiers, 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 3622*	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 3623*	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 3624*	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 3625*	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	2rspcedvdw 3626*	Double application of rspcedvdw 3616. (Contributed by SN, 24-Aug-2024.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)
Theorem	rspc2dv 3627*	2-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 6-Mar-2025.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	rspc3v 3628*	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 3629*	3-variable restricted existential 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	rspc3dv 3630*	3-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2025.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑧 = 𝐶 → (𝜏 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 𝜓)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐹)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	rspc4v 3631*	4-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2025.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))
Theorem	rspc6v 3632*	6-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑝 = 𝐸 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑞 = 𝐹 → (𝜁 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) ∧ (𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 𝜑 → 𝜓))
Theorem	rspc8v 3633*	8-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑝 = 𝐸 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑞 = 𝐹 → (𝜁 ↔ 𝜎))    &   ⊢ (𝑟 = 𝐺 → (𝜎 ↔ 𝜌))    &   ⊢ (𝑠 = 𝐻 → (𝜌 ↔ 𝜓))    ⇒   ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) ∧ ((𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) ∧ (𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌))) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜑 → 𝜓))
Theorem	rspceeqv 3634*	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	ralxpxfr2d 3635*	Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
⊢ 𝐴 ∈ V    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))
Theorem	rexraleqim 3636*	Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)
⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜑))    &   ⊢ (𝑧 = 𝑌 → (𝜑 ↔ 𝜃))    ⇒   ⊢ ((∃𝑧 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → 𝜃)
Theorem	eqvincg 3637*	A variable introduction law for class equality, closed form. (Contributed by Thierry Arnoux, 2-Mar-2017.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
Theorem	eqvinc 3638*	A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Thierry Arnoux, 23-Jan-2022.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))
Theorem	eqvincf 3639	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	alexeqg 3640*	Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. This is the analogue for classes of sbalex 2236. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
Theorem	ceqex 3641*	Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
Theorem	ceqsexg 3642*	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 3643*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) Drop ax-10 2138 and ax-12 2172. (Revised by Gino Giotto, 1-Dec-2023.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))
Theorem	ceqsrexv 3644*	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 3645*	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	ceqsralbv 3646*	Elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))
Theorem	ceqsrex2v 3647*	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	clel2g 3648*	Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent. (Revised by BJ, 12-Feb-2022.) Avoid ax-12 2172. (Revised by BJ, 1-Sep-2024.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))
Theorem	clel2gOLD 3649*	Obsolete version of clel2g 3648 as of 1-Sep-2024. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))
Theorem	clel2 3650*	Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
Theorem	clel3g 3651*	Alternate definition of membership in a set. (Contributed by NM, 13-Aug-2005.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))
Theorem	clel3 3652*	Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))
Theorem	clel4g 3653*	Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 2172. (Revised by BJ, 1-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)))
Theorem	clel4 3654*	Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))
Theorem	clel4OLD 3655*	Obsolete version of clel4 3654 as of 1-Sep-2024. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))
Theorem	clel5 3656*	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.) Remove use of ax-10 2138, ax-11 2155, and ax-12 2172. (Revised by Steven Nguyen, 19-May-2023.)
⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑋 = 𝑥)
Theorem	pm13.183 3657*	Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only 𝐴 is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.) Avoid ax-13 2372. (Revised by Wolf Lammen, 29-Apr-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
Theorem	rr19.3v 3658*	Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 4499 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rr19.28v 3659*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4501 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	elab6g 3660*	Membership in a class abstraction. Class version of sb6 2089. (Contributed by SN, 5-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	elabd2 3661*	Membership in a class abstraction, using implicit substitution. Deduction version of elab 3669. (Contributed by Gino Giotto, 12-Oct-2024.) (Revised by BJ, 16-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = {𝑥 ∣ 𝜓})    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒))
Theorem	elabd3 3662*	Membership in a class abstraction, using implicit substitution. Deduction version of elab 3669. (Contributed by Gino Giotto, 12-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒))
Theorem	elabgt 3663*	Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3667.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elabgtOLD 3664*	Obsolete version of elabgt 3663 as of 12-Oct-2024. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elabgf 3665	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 3666*	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	elabg 3667*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2372. (Revised by SN, 23-Nov-2022.) Avoid ax-10 2138, ax-11 2155, ax-12 2172. (Revised by SN, 5-Oct-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elabgOLD 3668*	Obsolete version of elabg 3667 as of 5-Oct-2024. (Contributed by NM, 14-Apr-1995.) Remove dependency on ax-13 2372. (Revised by SN, 23-Nov-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab 3669*	Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) Avoid ax-10 2138, ax-11 2155, ax-12 2172. (Revised by SN, 5-Oct-2024.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elabOLD 3670*	Obsolete version of elab 3669 as of 5-Oct-2024. (Contributed by NM, 1-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elab2g 3671*	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	elabd 3672*	Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝐴. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 23-Mar-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
Theorem	elab2 3673*	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 3674*	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 3675	Membership in a class abstraction, with a weaker antecedent than elabgf 3665. (Contributed by NM, 6-Sep-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab3g 3676*	Membership in a class abstraction, with a weaker antecedent than elabg 3667. (Contributed by NM, 29-Aug-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
Theorem	elab3 3677*	Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) (Revised by AV, 16-Aug-2024.)
⊢ (𝜓 → 𝐴 ∈ 𝑉)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
Theorem	elrabi 3678*	Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) Remove disjoint variable condition on 𝐴, 𝑥 and avoid ax-10 2138, ax-11 2155, ax-12 2172. (Revised by SN, 5-Aug-2024.)
⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)
Theorem	elrabiOLD 3679*	Obsolete version of elrabi 3678 as of 5-Aug-2024. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)
Theorem	elrabf 3680	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	rabtru 3681	Abstract builder using the constant wff ⊤. (Contributed by Thierry Arnoux, 4-May-2020.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴
Theorem	rabeqcOLD 3682*	Obsolete version of rabeqc 3445 as of 15-Jan-2025. (Contributed by AV, 20-Apr-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴
Theorem	elrab3t 3683*	Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3685.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
⊢ ((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓))
Theorem	elrab 3684*	Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) Remove dependency on ax-13 2372. (Revised by Steven Nguyen, 23-Nov-2022.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	elrab3 3685*	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 3686*	Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 3684. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	elrab2 3687*	Membership in a restricted 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 3688*	Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))
Theorem	ralabOLD 3689*	Obsolete version of ralab 3688 as of 2-Nov-2024. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))
Theorem	ralrab 3690*	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 3691*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))
Theorem	rexabOLD 3692*	Obsolete version of rexab 3691 as of 2-Nov-2024. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))
Theorem	rexrab 3693*	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 3694*	Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2109. (Revised by Gino Giotto, 1-Dec-2023.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒))
Theorem	ralrab2 3695*	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 3696*	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2109. (Revised by Gino Giotto, 1-Dec-2023.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒))
Theorem	rexrab2 3697*	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	reurab 3698*	Restricted existential uniqueness of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒))
Theorem	abidnf 3699*	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 3700*	A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 2149 and nfab 2910 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 3699 is useful. (Contributed by NM, 8-Dec-2006.)
⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ (Ⅎ𝑥𝐴 → 𝜑)