P. 35 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rabbii 3401	Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 3403. (Contributed by Peter Mazsa, 1-Nov-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabbidva 3402*	Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.) (Proof shortened by SN, 3-Dec-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabbidv 3403*	Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabbieq 3404	Equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 8-Jul-2019.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabswap 3405	Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐴}
Theorem	cbvrabv 3406*	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) Require 𝑥, 𝑦 be disjoint to avoid ax-11 2162 and ax-13 2374. (Revised by Steven Nguyen, 4-Dec-2022.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	rabeqcda 3407*	When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3408. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)
Theorem	rabeqc 3408*	A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.) (Proof shortened by SN, 15-Jan-2025.)
⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴
Theorem	rabeqi 3409	Equality theorem for restricted class abstractions. Inference form of rabeqf 3430. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2146, ax-11 2162, ax-12 2182. (Revised by GG, 3-Jun-2024.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}
Theorem	rabeq 3410*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) Avoid ax-10 2146, ax-11 2162, ax-12 2182. (Revised by GG, 20-Aug-2023.)
⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	rabeqdv 3411*	Equality of restricted class abstractions. Deduction form of rabeq 3410. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	rabeqbidva 3412*	Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	rabeqbidvaOLD 3413*	Obsolete version of rabeqbidva 3412 as of 1-Sep-2025. (Contributed by Mario Carneiro, 26-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	rabeqbidv 3414*	Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	rabrabi 3415*	Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2146, ax-11 2162 and ax-12 2182. (Revised by GG, 12-Oct-2024.)
⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))    ⇒   ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}
Theorem	nfrab1 3416	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	rabid 3417	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	rabidim1 3418	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
Theorem	reqabi 3419	Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑}    ⇒   ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
Theorem	rabrab 3420	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	rabbida4 3421	Version of rabbidva2 3398 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	rabbida 3422	Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3402 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2146, ax-11 2162. (Revised by Wolf Lammen, 14-Mar-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabbid 3423	Version of rabbidv 3403 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabeqd 3424	Deduction form of rabeq 3410. Note that contrary to rabeq 3410 it has no disjoint variable condition. (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	rabeqbida 3425	Version of rabeqbidva 3412 with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	rabbi 3426	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii 3401. (Contributed by NM, 25-Nov-2013.)
⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabid2f 3427	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	rabid2im 3428*	One direction of rabid2 3429 is based on fewer axioms. (Contributed by Wolf Lammen, 26-May-2025.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	rabid2 3429*	An "identity" law for restricted class abstraction. Prefer rabid2im 3428 if one direction is sufficient. (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	rabeqf 3430	Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	cbvrabw 3431*	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3436 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2374. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2146. (Revised by Wolf Lammen, 19-Jul-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	cbvrabwOLD 3432*	Obsolete version of cbvrabw 3431 as of 19-Jul-2025. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2374. (Revised by GG, 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	nfrabw 3433*	A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3435 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 13-Oct-2003.) Avoid ax-13 2374. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	rabbidaOLD 3434	Obsolete version of rabbida 3422 as of 14-Mar-2025. (Contributed by BJ, 27-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	nfrab 3435	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 2374. Use the weaker nfrabw 3433 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	cbvrab 3436	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker cbvrabw 3431 when possible. (Contributed by Andrew Salmon, 11-Jul-2011.) (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}
2.1.6  The universal class
Syntax	cvv 3437	Extend class notation to include the universal class symbol.
class V
Theorem	vjust 3438	Justification theorem for df-v 3439. (Contributed by Rodolfo Medina, 27-Apr-2010.)
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}
Definition	df-v 3439	Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. The class V can be described as the "class of all sets"; vprc 5255 proves that V is not itself a set in ZF. We will frequently use the expression 𝐴 ∈ V as a short way to say "𝐴 is a set", and isset 3451 proves that this expression has the same meaning as ∃𝑥𝑥 = 𝐴. In well-founded set theories without urelements, like ZF, the class V is equal to the von Neumann universe. However, the letter "V" does not stand for "von Neumann". The letter "V" was used earlier by Peano in 1889 for the universe of sets, where the letter V is derived from the Latin word "Verum", referring to the true truth constant 𝑇. Peano's notation V was adopted by Whitehead and Russell in Principia Mathematica for the class of all sets in 1910. The class constant V is the first class constant introduced in this database. As a constant, as opposed to a variable, it cannot be substituted with anything, and in particular it is not part of any disjoint variable condition. For a general discussion of the theory of classes, see mmset.html#class 3451. See dfv2 3440 for an alternate definition. (Contributed by NM, 26-May-1993.)
⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
Theorem	dfv2 3440	Alternate definition of the universal class (see df-v 3439). (Contributed by BJ, 30-Nov-2019.)
⊢ V = {𝑥 ∣ ⊤}
Theorem	vex 3441	All setvar variables are sets (see isset 3451). Theorem 6.8 of [Quine] p. 43. A shorter proof is possible from eleq2i 2825 but it uses more axioms. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2182. (Revised by SN, 28-Aug-2023.) (Proof shortened by BJ, 4-Sep-2024.)
⊢ 𝑥 ∈ V
Theorem	elv 3442	If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3441), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ (𝑥 ∈ V → 𝜑)    ⇒   ⊢ 𝜑
Theorem	elvd 3443	If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3441) and another antecedent, then it is implied by that other antecedent. Deduction associated with elv 3442. (Contributed by Peter Mazsa, 23-Oct-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	el2v 3444	If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3441), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	el3v 3445	If a proposition is implied by 𝑥 ∈ V, 𝑦 ∈ V and 𝑧 ∈ V (which is true, see vex 3441), then it is true. Inference forms (with ⊢ 𝐴 ∈ V, ⊢ 𝐵 ∈ V and ⊢ 𝐶 ∈ V hypotheses) of the general theorems (proving ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	el3v3 3446	If a proposition is implied by 𝑧 ∈ V (which is true, see vex 3441) and two other antecedents, then it is implied by these other antecedents. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	eqv 3447*	The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2146, ax-11 2162, ax-13 2374. (Revised by BJ, 10-Aug-2022.)
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	eqvf 3448	The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	abv 3449	The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 36971) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2808, ax-8 2115. (Revised by GG, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
Theorem	abvALT 3450	Alternate proof of abv 3449, shorter but using more axioms. (Contributed by BJ, 19-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
Theorem	isset 3451*	Two ways to express that "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3439) if and only if the class 𝐴 exists (i.e., there exists some set 𝑥 equal to class 𝐴). Theorem 6.9 of [Quine] p. 43. A class 𝐴 which is not a set is called a proper class. Conventions: We will often use the expression "𝐴 ∈ V " to mean "𝐴 is a set", for example in uniex 7680. To make some theorems more readily applicable, we will also use the more general expression 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to mean "𝐴 is a set", typically in an antecedent, or in a hypothesis for theorems in deduction form (see for instance uniexg 7679 compared with uniex 7680). That this is more general is seen either by substitution (when the variable 𝑉 has no other occurrences), or by elex 3458. (Contributed by NM, 26-May-1993.)
⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Theorem	cbvexeqsetf 3452*	The expression ∃𝑥𝑥 = 𝐴 means "𝐴 is a set" even if 𝐴 contains 𝑥 as a bound variable. This lemma helps minimizing axiom or df-clab 2712 usage in some cases. Extracted from the proof of issetft 3453. (Contributed by Wolf Lammen, 30-Jul-2025.)
⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
Theorem	issetft 3453	Closed theorem form of isset 3451 that does not require 𝑥 and 𝐴 to be distinct. Extracted from the proof of vtoclgft 3506. (Contributed by Wolf Lammen, 9-Apr-2025.)
⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
Theorem	issetf 3454	A version of isset 3451 that does not require 𝑥 and 𝐴 to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Theorem	isseti 3455*	A way to say "𝐴 is a set" (inference form). (Contributed by NM, 24-Jun-1993.) Remove dependencies on axioms. (Revised by BJ, 13-Jul-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 𝑥 = 𝐴
Theorem	issetri 3456*	A way to say "𝐴 is a set" (inference form). (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥 𝑥 = 𝐴    ⇒   ⊢ 𝐴 ∈ V
Theorem	eqvisset 3457	A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3451 and issetri 3456. (Contributed by BJ, 27-Apr-2019.)
⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
Theorem	elex 3458	If a class is a member of another class, then it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 28-May-2025.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
Theorem	elexOLD 3459	Obsolete version of elex 3458 as of 28-May-2025. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
Theorem	elexi 3460	If a class is a member of another class, then it is a set. Inference associated with elex 3458. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	elexd 3461	If a class is a member of another class, then it is a set. Deduction associated with elex 3458. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	elex22 3462*	If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
Theorem	prcnel 3463	A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by AV, 14-Nov-2020.)
⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉)
Theorem	ralv 3464	A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Theorem	rexv 3465	An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
Theorem	reuv 3466	A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
Theorem	rmov 3467	An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)
Theorem	rabab 3468	A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑}
Theorem	rexcom4b 3469*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)
Theorem	ceqsal1t 3470	One direction of ceqsalt 3471 is based on fewer assumptions and fewer axioms. It is at the same time the reverse direction of vtoclgft 3506. Extracted from a proof of ceqsalt 3471. (Contributed by Wolf Lammen, 25-Mar-2025.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	ceqsalt 3471*	Closed theorem version of ceqsalg 3473. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	ceqsralt 3472*	Restricted quantifier version of ceqsalt 3471. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	ceqsalg 3473*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3474. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	ceqsalgALT 3474*	Alternate proof of ceqsalg 3473, not using ceqsalt 3471. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	ceqsal 3475*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid df-clab 2712. (Revised by Wolf Lammen, 23-Jan-2025.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	ceqsalALT 3476*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. Shorter proof uses df-clab 2712. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	ceqsalv 3477*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid ax-12 2182. (Revised by SN, 8-Sep-2024.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	ceqsralv 3478*	Restricted quantifier version of ceqsalv 3477. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2123, ax-12 2182, ax-ext 2705. (Revised by SN, 8-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	gencl 3479*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝐵))    &   ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (𝜃 → 𝜓)
Theorem	2gencl 3480*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
⊢ (𝐶 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐶)    &   ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐷)    &   ⊢ (𝐴 = 𝐶 → (𝜑 ↔ 𝜓))    &   ⊢ (𝐵 = 𝐷 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝜑)    ⇒   ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝜒)
Theorem	3gencl 3481*	Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐷)    &   ⊢ (𝐹 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐹)    &   ⊢ (𝐺 ∈ 𝑆 ↔ ∃𝑧 ∈ 𝑅 𝐶 = 𝐺)    &   ⊢ (𝐴 = 𝐷 → (𝜑 ↔ 𝜓))    &   ⊢ (𝐵 = 𝐹 → (𝜓 ↔ 𝜒))    &   ⊢ (𝐶 = 𝐺 → (𝜒 ↔ 𝜃))    &   ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅) → 𝜑)    ⇒   ⊢ ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝜃)
Theorem	cgsexg 3482*	Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
⊢ (𝑥 = 𝐴 → 𝜒)    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓))
Theorem	cgsex2g 3483*	Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒)    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓))
Theorem	cgsex4g 3484*	An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2146, ax-11 2162. (Revised by GG, 28-Jun-2024.) Avoid ax-9 2123, ax-ext 2705. (Revised by Wolf Lammen, 21-Mar-2025.)
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))
Theorem	cgsex4gOLD 3485*	Obsolete version of cgsex4g 3484 as of 21-Mar-2025. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2146, ax-11 2162. (Revised by GG, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)    &   ⊢ (𝜒 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))
Theorem	ceqsex 3486*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Jan-2025.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
Theorem	ceqsexv 3487*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2182. (Revised by GG, 12-Oct-2024.) (Proof shortened by Wolf Lammen, 22-Jan-2025.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
Theorem	ceqsexv2d 3488*	Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025.) (Proof shortened by SN, 5-Jun-2025.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	ceqsexv2dOLD 3489*	Obsolete version of ceqsexv2d 3488 as of 5-Jun-2025. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	ceqsex2 3490*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜒    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)
Theorem	ceqsex2v 3491*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) Avoid ax-10 2146 and ax-11 2162. (Revised by GG, 20-Aug-2023.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)
Theorem	ceqsex3v 3492*	Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)
Theorem	ceqsex4v 3493*	Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏)
Theorem	ceqsex6v 3494*	Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁))    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁)
Theorem	ceqsex8v 3495*	Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    &   ⊢ 𝐺 ∈ V    &   ⊢ 𝐻 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑡 = 𝐺 → (𝜁 ↔ 𝜎))    &   ⊢ (𝑠 = 𝐻 → (𝜎 ↔ 𝜌))    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)
Theorem	gencbvex 3496*	Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ 𝐴 ∈ V    &   ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))    ⇒   ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))
Theorem	gencbvex2 3497*	Restatement of gencbvex 3496 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
⊢ 𝐴 ∈ V    &   ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜃 → ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))    ⇒   ⊢ (∃𝑥(𝜒 ∧ 𝜑) ↔ ∃𝑦(𝜃 ∧ 𝜓))
Theorem	gencbval 3498*	Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
⊢ 𝐴 ∈ V    &   ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))    ⇒   ⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓))
Theorem	sbhypf 3499*	Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3874. (Contributed by Raph Levien, 10-Apr-2004.) (Proof shortened by Wolf Lammen, 25-Jan-2025.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
Theorem	spcimgft 3500	Closed theorem form of spcimgf 3504. (Contributed by Wolf Lammen, 28-Jul-2025.)
⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))