P. 32 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3101-3200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reubida 3101	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	reubidva 3102*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	reubidv 3103*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
Theorem	reubiia 3104	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓)
Theorem	reubii 3105	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓)
Theorem	rmobida 3106	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	rmobidva 3107*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	rmobidv 3108*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
Theorem	rmobiia 3109	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)
Theorem	rmobii 3110	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐴 𝜓)
Theorem	raleqf 3111	Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	rexeqf 3112	Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
Theorem	reueq1f 3113	Equality theorem for restricted uniqueness 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	rmoeq1f 3114	Equality theorem for restricted uniqueness 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	raleq 3115*	Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	rexeq 3116*	Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
Theorem	reueq1 3117*	Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
Theorem	rmoeq1 3118*	Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	raleqi 3119*	Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)
Theorem	rexeqi 3120*	Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)
Theorem	raleqdv 3121*	Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	rexeqdv 3122*	Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
Theorem	raleqbi1dv 3123*	Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	rexeqbi1dv 3124*	Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓))
Theorem	reueqd 3125*	Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
Theorem	rmoeqd 3126*	Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓))
Theorem	raleqbid 3127	Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexeqbid 3128	Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	raleqbidv 3129*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexeqbidv 3130*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	raleqbidva 3131*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexeqbidva 3132*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	raleleq 3133*	All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.)
⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	raleleqALT 3134*	Alternate proof of raleleq 3133 using ralel 2907, being longer and using more axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	mormo 3135	Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	reu5 3136	Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
Theorem	reurex 3137	Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	reurmo 3138	Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	rmo5 3139	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 3140	Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	cbvralf 3141	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrexf 3142	Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvral 3143*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrex 3144*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreu 3145*	Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmo 3146*	Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvralv 3147*	Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrexv 3148*	Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
Theorem	cbvreuv 3149*	Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
Theorem	cbvrmov 3150*	Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
Theorem	cbvraldva2 3151*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒))
Theorem	cbvrexdva2 3152*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
Theorem	cbvraldva 3153*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))
Theorem	cbvrexdva 3154*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒))
Theorem	cbvral2v 3155*	Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
Theorem	cbvrex2v 3156*	Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
Theorem	cbvral3v 3157*	Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑣 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑧 = 𝑢 → (𝜃 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
Theorem	cbvralsv 3158*	Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
Theorem	cbvrexsv 3159*	Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
Theorem	sbralie 3160*	Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)
Theorem	rabbiia 3161	Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabbidva2 3162*	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	rabbidva 3163*	Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabbidv 3164*	Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
Theorem	rabeqf 3165	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	rabeq 3166*	Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑})
Theorem	rabeqdv 3167*	Equality of restricted class abstractions. Deduction form of rabeq 3166. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	rabeqbidv 3168*	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	rabeqbidva 3169*	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	rabeq2i 3170	Inference rule 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	cbvrab 3171	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. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
Theorem	cbvrabv 3172*	Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
2.1.6  The universal class
Syntax	cvv 3173	Extend class notation to include the universal class symbol.
class V
Theorem	vjust 3174	Soundness justification theorem for df-v 3175. (Contributed by Rodolfo Medina, 27-Apr-2010.)
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}
Definition	df-v 3175	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 4724 proves that V is not itself a set in ZFC. We will frequently use the expression 𝐴 ∈ V as a short way to say "𝐴 is a set", and isset 3180 proves that this expression has the same meaning as ∃𝑥𝑥 = 𝐴. The class V is called the "von Neumann universe", however, the letter "V" is not a tribute to the name of 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 word "Verum". Peano's notation V was adopted by Whitehead and Russell in Principia Mathematica for the class of all sets in 1910. For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)
⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
Theorem	vex 3176	All setvar variables are sets (see isset 3180). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.)
⊢ 𝑥 ∈ V
Theorem	eqvf 3177	The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	eqv 3178*	The universe contains every set. (Contributed by NM, 11-Sep-2006.)
⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	abv 3179	The class of sets verifying a property is the universal class if and only if that property is a tautology. (Contributed by BJ, 19-Mar-2021.)
⊢ ({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
Theorem	isset 3180*	Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3175) 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. Notational convention: We will use the notational device "𝐴 ∈ V " to mean "𝐴 is a set" very frequently, for example in uniex 6851. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 6853, in order to shorten certain proofs we use the more general antecedent 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to mean "𝐴 is a set." Note that a constant is implicitly considered distinct from all variables. This is why V is not included in the distinct variable list, even though df-clel 2606 requires that the expression substituted for 𝐵 not contain 𝑥. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.)
⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Theorem	issetf 3181	A version of isset 3180 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 3182*	A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 24-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 𝑥 = 𝐴
Theorem	issetri 3183*	A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥 𝑥 = 𝐴    ⇒   ⊢ 𝐴 ∈ V
Theorem	eqvisset 3184	A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 3180 and issetri 3183. (Contributed by BJ, 27-Apr-2019.)
⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
Theorem	elex 3185	If a class is a member of another class, 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 3186	If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	elexd 3187	If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	elisset 3188*	An element of a class exists. (Contributed by NM, 1-May-1995.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
Theorem	elex2 3189*	If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)
Theorem	elex22 3190*	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 3191	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 3192	A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
Theorem	rexv 3193	An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
Theorem	reuv 3194	A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
Theorem	rmov 3195	A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)
Theorem	rabab 3196	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	ralcom4 3197*	Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexcom4 3198*	Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
Theorem	rexcom4a 3199*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓))
Theorem	rexcom4b 3200*	Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)