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	nfrmod 3401	Deduction version of nfrmo 3403. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓)
Theorem	nfreud 3402	Deduction version of nfreu 3404. Usage of this theorem is discouraged because it depends on ax-13 2370. (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	nfrmo 3403	Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfrmow 3384 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	nfreu 3404	Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfreuw 3385 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
2.1.5.3  Restricted class abstraction
Syntax	crab 3405	Extend class notation to include the restricted class abstraction (class builder).
class {𝑥 ∈ 𝐴 ∣ 𝜑}
Definition	df-rab 3406	Define a restricted class abstraction (class builder): {𝑥 ∈ 𝐴 ∣ 𝜑} is the class of all sets 𝑥 in 𝐴 such that 𝜑(𝑥) is true. Definition of [TakeutiZaring] p. 20. For the interpretation given in the previous paragraph to be correct, we need to assume Ⅎ𝑥𝐴, which is the case as soon as 𝑥 and 𝐴 are disjoint, which is generally the case. If 𝐴 were to depend on 𝑥, then the interpretation would be less obvious (think of the two extreme cases 𝐴 = {𝑥} and 𝐴 = 𝑥, for instance). See also df-ral 3061. (Contributed by NM, 22-Nov-1994.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
Theorem	rabbidva2 3407*	Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	rabbia2 3408	Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}
Theorem	rabbiia 3409	Equivalent formulas yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.) (Proof shortened by Wolf Lammen, 12-Jan-2025.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabbiiaOLD 3410	Obsolete version of rabbiia 3409 as of 12-Jan-2025. (Contributed by NM, 22-May-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabbii 3411	Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 3413. (Contributed by Peter Mazsa, 1-Nov-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabbidva 3412*	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 3413*	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	rabswap 3414	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 3415*	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 2154 and ax-13 2370. (Revised by Steven Nguyen, 4-Dec-2022.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	rabeqcda 3416*	When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3417. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)
Theorem	rabeqc 3417*	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 3418	Equality theorem for restricted class abstractions. Inference form of rabeqf 3439. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2137, ax-11 2154, ax-12 2171. (Revised by Gino Giotto, 3-Jun-2024.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}
Theorem	rabeq 3419*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) Avoid ax-10 2137, ax-11 2154, ax-12 2171. (Revised by Gino Giotto, 20-Aug-2023.)
⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	rabeqdv 3420*	Equality of restricted class abstractions. Deduction form of rabeq 3419. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	rabeqbidva 3421*	Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
Theorem	rabeqbidv 3422*	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 3423*	Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2137, ax-11 2154 and ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))    ⇒   ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}
Theorem	nfrab1 3424	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	rabid 3425	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 3426	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
Theorem	reqabi 3427	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 3428	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	rabrabiOLD 3429*	Obsolete version of rabrabi 3423 as of 12-Oct-2024. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2137 and ax-11 2154. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))    ⇒   ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}
Theorem	rabbi 3430	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii 3411. (Contributed by NM, 25-Nov-2013.)
⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabbida 3431	Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3412 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabbid 3432	Version of rabbidv 3413 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	rabbida4 3433	Version of rabbidva2 3407 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	rabeqd 3434	Deduction form of rabeq 3419. Note that contrary to rabeq 3419 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 3435	Version of rabeqbidva 3421 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	rabid2f 3436	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	rabid2 3437*	An "identity" law for restricted class abstraction. (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	rabid2OLD 3438*	Obsolete version of rabid2 3437 as of 24-11-2024. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rabeqf 3439	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 3440*	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3445 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	nfrabw 3441*	A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3444 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 13-Oct-2003.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	nfrabwOLD 3442*	Obsolete version of nfrabw 3441 as of 23-Nov2024. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	rabeqiOLD 3443	Obsolete version of rabeqi 3418 as of 3-Jun-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2137 and ax-11 2154. (Revised by Gino Giotto, 20-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}
Theorem	nfrab 3444	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 2370. Use the weaker nfrabw 3441 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 3445	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 2370. Use the weaker cbvrabw 3440 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 3446	Extend class notation to include the universal class symbol.
class V
Theorem	vjust 3447	Justification theorem for df-v 3448. (Contributed by Rodolfo Medina, 27-Apr-2010.)
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}
Definition	df-v 3448	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 5277 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 3459 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 3459. See dfv2 3449 for an alternate definition. (Contributed by NM, 26-May-1993.)
⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
Theorem	dfv2 3449	Alternate definition of the universal class (see df-v 3448). (Contributed by BJ, 30-Nov-2019.)
⊢ V = {𝑥 ∣ ⊤}
Theorem	vex 3450	All setvar variables are sets (see isset 3459). Theorem 6.8 of [Quine] p. 43. A shorter proof is possible from eleq2i 2824 but it uses more axioms. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2171. (Revised by SN, 28-Aug-2023.) (Proof shortened by BJ, 4-Sep-2024.)
⊢ 𝑥 ∈ V
Theorem	vexOLD 3451	Obsolete version of vex 3450 as of 4-Sep-2024. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2171. (Revised by SN, 28-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑥 ∈ V
Theorem	elv 3452	If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3450), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ (𝑥 ∈ V → 𝜑)    ⇒   ⊢ 𝜑
Theorem	elvd 3453	If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3450) and another antecedent, then it is implied by that other antecedent. Deduction associated with elv 3452. (Contributed by Peter Mazsa, 23-Oct-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	el2v 3454	If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3450), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	eqv 3455*	The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2137, ax-11 2154, ax-13 2370. (Revised by BJ, 10-Aug-2022.)
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	eqvf 3456	The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	abv 3457	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 35449) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2809, ax-8 2108. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
Theorem	abvALT 3458	Alternate proof of abv 3457, 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 3459*	Two ways to express that "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3448) 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 7683. 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 7682 compared with uniex 7683). That this is more general is seen either by substitution (when the variable 𝑉 has no other occurrences), or by elex 3464. (Contributed by NM, 26-May-1993.)
⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Theorem	issetf 3460	A version of isset 3459 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 3461*	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 3462*	A way to say "𝐴 is a set" (inference form). (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥 𝑥 = 𝐴    ⇒   ⊢ 𝐴 ∈ V
Theorem	eqvisset 3463	A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3459 and issetri 3462. (Contributed by BJ, 27-Apr-2019.)
⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
Theorem	elex 3464	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.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
Theorem	elexi 3465	If a class is a member of another class, then it is a set. Inference associated with elex 3464. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	elexd 3466	If a class is a member of another class, then it is a set. Deduction associated with elex 3464. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	elex2OLD 3467*	Obsolete version of elex2 2811 as of 30-Nov-2024. (Contributed by Alan Sare, 25-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)
Theorem	elex22 3468*	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 3469	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 3470	A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Theorem	rexv 3471	An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
Theorem	reuv 3472	A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
Theorem	rmov 3473	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 3474	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 3475*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)
Theorem	ceqsalt 3476*	Closed theorem version of ceqsalg 3478. (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 3477*	Restricted quantifier version of ceqsalt 3476. (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 3478*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3479. (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 3479*	Alternate proof of ceqsalg 3478, not using ceqsalt 3476. (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 3480*	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 2709. (Revised by Wolf Lammen, 23-Jan-2025.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	ceqsalALT 3481*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. Shorter proof uses df-clab 2709. (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 3482*	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 2171. (Revised by SN, 8-Sep-2024.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	ceqsalvOLD 3483*	Obsolete version of ceqsalv 3482 as of 8-Sep-2024. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	ceqsralv 3484*	Restricted quantifier version of ceqsalv 3482. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2116, ax-12 2171, ax-ext 2702. (Revised by SN, 8-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	ceqsralvOLD 3485*	Obsolete version of ceqsalv 3482 as of 8-Sep-2024. (Contributed by NM, 21-Jun-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	gencl 3486*	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 3487*	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 3488*	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 3489*	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 3490*	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 3491*	An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2137, ax-11 2154. (Revised by Gino Giotto, 28-Jun-2024.)
⊢ (((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 3492*	Obsolete version of cgsex4g 3491 as of 28-Jun-2024. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2137. (Revised by Gino Giotto, 20-Aug-2023.) (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 3493*	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	ceqsexOLD 3494*	Obsolete version of ceqsex 3493 as of 22-Jan-2025. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
Theorem	ceqsexv 3495*	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.) (Proof shortened by Wolf Lammen, 22-Jan-2025.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
Theorem	ceqsexvOLD 3496*	Obsolete version of ceqsexv 3495 as of 12-Oct-2024. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
Theorem	ceqsexvOLDOLD 3497*	Obsolete version of ceqsexv 3495 as of 12-Oct-2024. (Contributed by NM, 2-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
Theorem	ceqsexv2d 3498*	Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	ceqsex2 3499*	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 3500*	Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) Avoid ax-10 2137 and ax-11 2154. (Revised by Gino Giotto, 20-Aug-2023.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)