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	reuanidOLD 3401	Obsolete version of reuanid 3399 as of 12-Jan-2025. (Contributed by Peter Mazsa, 12-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑)
Theorem	2reu2rex 3402	Double restricted existential uniqueness, analogous to 2eu2ex 2646. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	rmobidva 3403*	Formula-building rule for restricted at-most-one quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) Avoid ax-6 1967, ax-7 2007, ax-12 2178. (Revised by Wolf Lammen, 23-Nov-2024.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	reubidva 3404*	Formula-building rule for restricted existential uniqueness quantifier (deduction form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	rmobidv 3405*	Formula-building rule for restricted at-most-one quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	reubidv 3406*	Formula-building rule for restricted existential uniqueness quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	reueubd 3407*	Restricted existential uniqueness is equivalent to existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.)
⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝑉 𝜓 ↔ ∃!𝑥𝜓))
Theorem	rmo5 3408	Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
Theorem	nrexrmo 3409	Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	moel 3410*	"At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2158. (Revised by Wolf Lammen, 23-Nov-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)
Theorem	cbvrmovw 3411*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov 3437 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 16-Jun-2017.) (Revised by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuvw 3412*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3438 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004.) (Revised by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	moelOLD 3413*	Obsolete version of moel 3410 as of 23-Nov-2024. (Contributed by Thierry Arnoux, 26-Jul-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)
Theorem	rmobida 3414	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	reubida 3415	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by Mario Carneiro, 19-Nov-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	rmobidvaOLD 3416*	Obsolete version of rmobidv 3405 as of 23-Nov-2024. (Contributed by NM, 16-Jun-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	cbvrmow 3417*	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3436 with a disjoint variable condition, which does not require ax-10 2141, ax-13 2380. (Contributed by NM, 16-Jun-2017.) Avoid ax-10 2141, ax-13 2380. (Revised by GG, 23-May-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuw 3418*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3435 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2141. (Revised by Wolf Lammen, 10-Dec-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	nfrmo1 3419	The setvar 𝑥 is not free in ∃*𝑥 ∈ 𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝜑
Theorem	nfreu1 3420	The setvar 𝑥 is not free in ∃!𝑥 ∈ 𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜑
Theorem	nfrmow 3421*	Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3441 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 16-Jun-2017.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2118, ax-ext 2711. (Revised by Wolf Lammen, 21-Nov-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	nfreuw 3422*	Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3442 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2118, ax-ext 2711. (Revised by Wolf Lammen, 21-Nov-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
Theorem	cbvreuwOLD 3423*	Obsolete version of cbvreuw 3418 as of 10-Dec-2024. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by GG, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuvwOLD 3424*	Obsolete version of cbvreuvw 3412 as of 30-Sep-2024. (Contributed by NM, 5-Apr-2004.) (Revised by GG, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	rmoeq1 3425*	Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2141, ax-11 2158, and ax-12 2178. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2110. (Revised by Wolf Lammen, 12-Mar-2025.)
⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	reueq1 3426*	Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2141, ax-11 2158, and ax-12 2178. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2110. (Revised by Wolf Lammen, 12-Mar-2025.)
⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
Theorem	rmoeq1OLD 3427*	Obsolete version of rmoeq1 3425 as of 12-Mar-2025. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2141, ax-11 2158, and ax-12 2178. (Revised by Steven Nguyen, 30-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	reueq1OLD 3428*	Obsolete version of reueq1 3426 as of 12-Mar-2025. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2141, ax-11 2158, and ax-12 2178. (Revised by Steven Nguyen, 30-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
Theorem	rmoeqd 3429*	Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓))
Theorem	reueqd 3430*	Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
Theorem	rmoeq1f 3431	Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	reueq1f 3432	Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
Theorem	nfreuwOLD 3433*	Obsolete version of nfreuw 3422 as of 21-Nov-2024. (Contributed by NM, 30-Oct-2010.) (Revised by GG, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
Theorem	nfrmowOLD 3434*	Obsolete version of nfrmow 3421 as of 21-Nov-2024. (Contributed by NM, 16-Jun-2017.) (Revised by GG, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	cbvreu 3435*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker cbvreuw 3418 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmo 3436*	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 2380. Use the weaker cbvrmow 3417, cbvrmovw 3411 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 3437*	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 2380. (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 3438*	Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3412 for a version without ax-13 2380, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker cbvreuvw 3412 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 3439	Deduction version of nfrmo 3441. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓)
Theorem	nfreud 3440	Deduction version of nfreu 3442. Usage of this theorem is discouraged because it depends on ax-13 2380. (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 3441	Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfrmow 3421 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
Theorem	nfreu 3442	Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfreuw 3422 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 3443	Extend class notation to include the restricted class abstraction (class builder).
class {𝑥 ∈ 𝐴 ∣ 𝜑}
Definition	df-rab 3444	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 3068. (Contributed by NM, 22-Nov-1994.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
Theorem	rabbidva2 3445*	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 3446	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 3447	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 3448	Obsolete version of rabbiia 3447 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 3449	Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 3451. (Contributed by Peter Mazsa, 1-Nov-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabbidva 3450*	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 3451*	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 3452	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 3453	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 3454*	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 2380. (Revised by Steven Nguyen, 4-Dec-2022.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	rabeqcda 3455*	When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3456. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)
Theorem	rabeqc 3456*	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 3457	Equality theorem for restricted class abstractions. Inference form of rabeqf 3480. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2141, ax-11 2158, ax-12 2178. (Revised by GG, 3-Jun-2024.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}
Theorem	rabeq 3458*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) Avoid ax-10 2141, ax-11 2158, ax-12 2178. (Revised by GG, 20-Aug-2023.)
⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	rabeqdv 3459*	Equality of restricted class abstractions. Deduction form of rabeq 3458. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	rabeqbidva 3460*	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 3461*	Obsolete version of rabeqbidva 3460 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 3462*	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 3463*	Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2141, 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 3464	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	rabid 3465	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 3466	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
Theorem	reqabi 3467	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 3468	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 3469*	Obsolete version of rabrabi 3463 as of 12-Oct-2024. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2141 and ax-11 2158. (Revised by GG, 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	rabbida4 3470	Version of rabbidva2 3445 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 3471	Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3450 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2141, 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 3472	Version of rabbidv 3451 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 3473	Deduction form of rabeq 3458. Note that contrary to rabeq 3458 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 3474	Version of rabeqbidva 3460 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 3475	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii 3449. (Contributed by NM, 25-Nov-2013.)
⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabid2f 3476	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 3477*	One direction of rabid2 3478 is based on fewer axioms. (Contributed by Wolf Lammen, 26-May-2025.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑})
Theorem	rabid2 3478*	An "identity" law for restricted class abstraction. Prefer rabid2im 3477 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	rabid2OLD 3479*	Obsolete version of rabid2 3478 as of 24-Nov-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 3480	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 3481*	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3487 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2141. (Revised by Wolf Lammen, 19-Jul-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	cbvrabwOLD 3482*	Obsolete version of cbvrabw 3481 as of 19-Jul-2025. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2380. (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 3483*	A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3486 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 13-Oct-2003.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	nfrabwOLD 3484*	Obsolete version of nfrabw 3483 as of 23-Nov2024. (Contributed by NM, 13-Oct-2003.) (Revised by GG, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
Theorem	rabbidaOLD 3485	Obsolete version of rabbida 3471 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 3486	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 2380. Use the weaker nfrabw 3483 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 3487	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 2380. Use the weaker cbvrabw 3481 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 3488	Extend class notation to include the universal class symbol.
class V
Theorem	vjust 3489	Justification theorem for df-v 3490. (Contributed by Rodolfo Medina, 27-Apr-2010.)
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}
Definition	df-v 3490	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 5333 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 3502 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 3502. See dfv2 3491 for an alternate definition. (Contributed by NM, 26-May-1993.)
⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
Theorem	dfv2 3491	Alternate definition of the universal class (see df-v 3490). (Contributed by BJ, 30-Nov-2019.)
⊢ V = {𝑥 ∣ ⊤}
Theorem	vex 3492	All setvar variables are sets (see isset 3502). Theorem 6.8 of [Quine] p. 43. A shorter proof is possible from eleq2i 2836 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 3493	If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3492), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ (𝑥 ∈ V → 𝜑)    ⇒   ⊢ 𝜑
Theorem	elvd 3494	If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3492) and another antecedent, then it is implied by that other antecedent. Deduction associated with elv 3493. (Contributed by Peter Mazsa, 23-Oct-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	el2v 3495	If a proposition is implied by 𝑥 ∈ V and 𝑦 ∈ V (which is true, see vex 3492), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	el3v 3496	If a proposition is implied by 𝑥 ∈ V, 𝑦 ∈ V and 𝑧 ∈ V (which is true, see vex 3492), 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 3497	If a proposition is implied by 𝑧 ∈ V (which is true, see vex 3492) 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 3498*	The universe contains every set. (Contributed by NM, 11-Sep-2006.) Remove dependency on ax-10 2141, ax-11 2158, ax-13 2380. (Revised by BJ, 10-Aug-2022.)
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	eqvf 3499	The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	abv 3500	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 36872) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2819, ax-8 2110. (Revised by GG, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)