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	cbvrmo 3401*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvrmow 3383, cbvrmovw 3379 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmov 3402*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuv 3403*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3380 for a version without ax-13 2371, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvreuvw 3380 when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	nfrmod 3404	Deduction version of nfrmo 3406. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓)
Theorem	nfreud 3405	Deduction version of nfreu 3407. Usage of this theorem is discouraged because it depends on ax-13 2371. (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 3406	Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfrmow 3387 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	nfreu 3407	Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker nfreuw 3388 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 3408	Extend class notation to include the restricted class abstraction (class builder).
class {𝑥 ∈ 𝐴 ∣ 𝜑}
Definition	df-rab 3409	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 3046. (Contributed by NM, 22-Nov-1994.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
Theorem	rabbidva2 3410*	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 3411	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 3412	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 3413	Obsolete version of rabbiia 3412 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 3414	Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 3416. (Contributed by Peter Mazsa, 1-Nov-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabbidva 3415*	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 3416*	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 3417	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 3418	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 3419*	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 2158 and ax-13 2371. (Revised by Steven Nguyen, 4-Dec-2022.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	rabeqcda 3420*	When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3421. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)
Theorem	rabeqc 3421*	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 3422	Equality theorem for restricted class abstractions. Inference form of rabeqf 3443. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2142, ax-11 2158, ax-12 2178. (Revised by GG, 3-Jun-2024.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}
Theorem	rabeq 3423*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) Avoid ax-10 2142, ax-11 2158, ax-12 2178. (Revised by GG, 20-Aug-2023.)
⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	rabeqdv 3424*	Equality of restricted class abstractions. Deduction form of rabeq 3423. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	rabeqbidva 3425*	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 3426*	Obsolete version of rabeqbidva 3425 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 3427*	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 3428*	Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2142, ax-11 2158 and ax-12 2178. (Revised by GG, 12-Oct-2024.)
⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))    ⇒   ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}
Theorem	nfrab1 3429	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	rabid 3430	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 3431	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
Theorem	reqabi 3432	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 3433	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 3434	Version of rabbidva2 3410 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 3435	Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3415 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2142, ax-11 2158. (Revised by Wolf Lammen, 14-Mar-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabbid 3436	Version of rabbidv 3416 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 3437	Deduction form of rabeq 3423. Note that contrary to rabeq 3423 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 3438	Version of rabeqbidva 3425 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 3439	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii 3414. (Contributed by NM, 25-Nov-2013.)
⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabid2f 3440	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 3441*	One direction of rabid2 3442 is based on fewer axioms. (Contributed by Wolf Lammen, 26-May-2025.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	rabid2 3442*	An "identity" law for restricted class abstraction. Prefer rabid2im 3441 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 3443	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 3444*	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3449 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2371. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2142. (Revised by Wolf Lammen, 19-Jul-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	cbvrabwOLD 3445*	Obsolete version of cbvrabw 3444 as of 19-Jul-2025. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2371. (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 3446*	A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3448 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 13-Oct-2003.) Avoid ax-13 2371. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	rabbidaOLD 3447	Obsolete version of rabbida 3435 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 3448	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 2371. Use the weaker nfrabw 3446 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 3449	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 2371. Use the weaker cbvrabw 3444 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 3450	Extend class notation to include the universal class symbol.
class V
Theorem	vjust 3451	Justification theorem for df-v 3452. (Contributed by Rodolfo Medina, 27-Apr-2010.)
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}
Definition	df-v 3452	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 5273 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 3464 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 3464. See dfv2 3453 for an alternate definition. (Contributed by NM, 26-May-1993.)
⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
Theorem	dfv2 3453	Alternate definition of the universal class (see df-v 3452). (Contributed by BJ, 30-Nov-2019.)
⊢ V = {𝑥 ∣ ⊤}
Theorem	vex 3454	All setvar variables are sets (see isset 3464). Theorem 6.8 of [Quine] p. 43. A shorter proof is possible from eleq2i 2821 but it uses more axioms. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2178. (Revised by SN, 28-Aug-2023.) (Proof shortened by BJ, 4-Sep-2024.)
⊢ 𝑥 ∈ V
Theorem	elv 3455	If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3454), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ (𝑥 ∈ V → 𝜑)    ⇒   ⊢ 𝜑
Theorem	elvd 3456	If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3454) and another antecedent, then it is implied by that other antecedent. Deduction associated with elv 3455. (Contributed by Peter Mazsa, 23-Oct-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	el2v 3457	If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3454), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	el3v 3458	If a proposition is implied by 𝑥 ∈ V, 𝑦 ∈ V and 𝑧 ∈ V (which is true, see vex 3454), 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 3459	If a proposition is implied by 𝑧 ∈ V (which is true, see vex 3454) 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 3460*	The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2142, ax-11 2158, ax-13 2371. (Revised by BJ, 10-Aug-2022.)
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	eqvf 3461	The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	abv 3462	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 36901) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2804, ax-8 2111. (Revised by GG, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
Theorem	abvALT 3463	Alternate proof of abv 3462, 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 3464*	Two ways to express that "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3452) 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 7720. 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 7719 compared with uniex 7720). That this is more general is seen either by substitution (when the variable 𝑉 has no other occurrences), or by elex 3471. (Contributed by NM, 26-May-1993.)
⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Theorem	cbvexeqsetf 3465*	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 2709 usage in some cases. Extracted from the proof of issetft 3466. (Contributed by Wolf Lammen, 30-Jul-2025.)
⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
Theorem	issetft 3466	Closed theorem form of isset 3464 that does not require 𝑥 and 𝐴 to be distinct. Extracted from the proof of vtoclgft 3521. (Contributed by Wolf Lammen, 9-Apr-2025.)
⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
Theorem	issetf 3467	A version of isset 3464 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 3468*	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 3469*	A way to say "𝐴 is a set" (inference form). (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥 𝑥 = 𝐴    ⇒   ⊢ 𝐴 ∈ V
Theorem	eqvisset 3470	A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 3464 and issetri 3469. (Contributed by BJ, 27-Apr-2019.)
⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
Theorem	elex 3471	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 3472	Obsolete version of elex 3471 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 3473	If a class is a member of another class, then it is a set. Inference associated with elex 3471. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	elexd 3474	If a class is a member of another class, then it is a set. Deduction associated with elex 3471. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	elex22 3475*	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 3476	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 3477	A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Theorem	rexv 3478	An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
Theorem	reuv 3479	A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
Theorem	rmov 3480	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 3481	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 3482*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)
Theorem	ceqsal1t 3483	One direction of ceqsalt 3484 is based on fewer assumptions and fewer axioms. It is at the same time the reverse direction of vtoclgft 3521. Extracted from a proof of ceqsalt 3484. (Contributed by Wolf Lammen, 25-Mar-2025.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	ceqsalt 3484*	Closed theorem version of ceqsalg 3486. (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 3485*	Restricted quantifier version of ceqsalt 3484. (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 3486*	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3487. (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 3487*	Alternate proof of ceqsalg 3486, not using ceqsalt 3484. (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 3488*	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 3489*	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 3490*	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 2178. (Revised by SN, 8-Sep-2024.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
Theorem	ceqsralv 3491*	Restricted quantifier version of ceqsalv 3490. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2119, ax-12 2178, ax-ext 2702. (Revised by SN, 8-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
Theorem	gencl 3492*	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 3493*	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 3494*	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 3495*	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 3496*	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 3497*	An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2142, ax-11 2158. (Revised by GG, 28-Jun-2024.) Avoid ax-9 2119, ax-ext 2702. (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 3498*	Obsolete version of cgsex4g 3497 as of 21-Mar-2025. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2142, ax-11 2158. (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 3499*	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 3500*	Obsolete version of ceqsex 3499 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)