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Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrabbi 3001 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 3067. (Contributed by NM, 25-Nov-2013.)
(∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrabswap 3002 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
{𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}
 
Theoremnfrab1 3003 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
𝑥{𝑥𝐴𝜑}
 
Theoremnfrab 3004 A variable not free in a wff remains so in a restricted class abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.)
𝑥𝜑    &   𝑥𝐴       𝑥{𝑦𝐴𝜑}
 
Theoremreubida 3005 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubidva 3006* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubidv 3007* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
 
Theoremreubiia 3008 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
(𝑥𝐴 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 
Theoremreubii 3009 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
(𝜑𝜓)       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)
 
Theoremrmobida 3010 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobidva 3011* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobidv 3012* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))
 
Theoremrmobiia 3013 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
(𝑥𝐴 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
 
Theoremrmobii 3014 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
(𝜑𝜓)       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)
 
Theoremraleqf 3015 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
 
Theoremrexeqf 3016 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
 
Theoremreueq1f 3017 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
 
Theoremrmoeq1f 3018 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
 
Theoremraleq 3019* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
 
Theoremrexeq 3020* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
(𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
 
Theoremreueq1 3021* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
 
Theoremrmoeq1 3022* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
 
Theoremraleqi 3023* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑)
 
Theoremrexeqi 3024* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐴 = 𝐵       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)
 
Theoremraleqdv 3025* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
 
Theoremrexeqdv 3026* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))
 
Theoremraleqbi1dv 3027* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
 
Theoremrexeqbi1dv 3028* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))
 
Theoremreueqd 3029* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))
 
Theoremrmoeqd 3030* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜓))
 
Theoremraleqbid 3031 Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremrexeqbid 3032 Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremraleqbidv 3033* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremrexeqbidv 3034* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremraleqbidva 3035* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
 
Theoremrexeqbidva 3036* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))
 
Theoremraleleq 3037* All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.)
(𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)
 
TheoremraleleqALT 3038* Alternate proof of raleleq 3037 using ralel 2811, being longer and using more axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → ∀𝑥𝐴 𝑥𝐵)
 
Theoremmormo 3039 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝜑 → ∃*𝑥𝐴 𝜑)
 
Theoremreu5 3040 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
(∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐴 𝜑))
 
Theoremreurex 3041 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
(∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)
 
Theoremreurmo 3042 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
(∃!𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)
 
Theoremrmo5 3043 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
 
Theoremnrexrmo 3044 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
(¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)
 
Theoremcbvralf 3045 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
Theoremcbvrexf 3046 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
Theoremcbvral 3047* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
Theoremcbvrex 3048* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
Theoremcbvreu 3049* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 
Theoremcbvrmo 3050* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
 
Theoremcbvralv 3051* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
Theoremcbvrexv 3052* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
Theoremcbvreuv 3053* 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 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 
Theoremcbvrmov 3054* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
 
Theoremcbvraldva2 3055* 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.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒))
 
Theoremcbvrexdva2 3056* 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.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))    &   ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
 
Theoremcbvraldva 3057* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐴 𝜒))
 
Theoremcbvrexdva 3058* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐴 𝜒))
 
Theoremcbvral2v 3059* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
 
Theoremcbvrex2v 3060* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
 
Theoremcbvral3v 3061* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
(𝑥 = 𝑤 → (𝜑𝜒))    &   (𝑦 = 𝑣 → (𝜒𝜃))    &   (𝑧 = 𝑢 → (𝜃𝜓))       (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
 
Theoremcbvralsv 3062* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
 
Theoremcbvrexsv 3063* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
 
Theoremsbralie 3064* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
(𝑥 = 𝑦 → (𝜑𝜓))       ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ ∀𝑦𝑥 𝜓)
 
Theoremrabbiia 3065 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}
 
Theoremrabbidva2 3066* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabbidva 3067* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrabbidv 3068* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrabeqf 3069 Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
 
Theoremrabeq 3070* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
(𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
 
Theoremrabeqdv 3071* Equality of restricted class abstractions. Deduction form of rabeq 3070. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
 
Theoremrabeqbidv 3072* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabeqbidva 3073* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 
Theoremrabeq2i 3074 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴 ↔ (𝑥𝐵𝜑))
 
Theoremcbvrab 3075 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.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
 
Theoremcbvrabv 3076* Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
 
2.1.6  The universal class
 
Syntaxcvv 3077 Extend class notation to include the universal class symbol.
class V
 
Theoremvjust 3078 Soundness justification theorem for df-v 3079. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
 
Definitiondf-v 3079 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 4623 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 3084 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 = {𝑥𝑥 = 𝑥}
 
Theoremvex 3080 All setvar variables are sets (see isset 3084). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.)
𝑥 ∈ V
 
Theoremeqvf 3081 The universe contains every set. (Contributed by BJ, 15-Jul-2021.)
𝑥𝐴       (𝐴 = V ↔ ∀𝑥 𝑥𝐴)
 
Theoremeqv 3082* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
(𝐴 = V ↔ ∀𝑥 𝑥𝐴)
 
Theoremabv 3083 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.)
({𝑥𝜑} = V ↔ ∀𝑥𝜑)
 
Theoremisset 3084* Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3079) 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 6727. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 6729, in order to shorten certain proofs we use the more general antecedent 𝐴𝑉 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 2510 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)
 
Theoremissetf 3085 A version of isset 3084 that does not require 𝑥 and 𝐴 to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
 
Theoremisseti 3086* A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 24-Jun-1993.)
𝐴 ∈ V       𝑥 𝑥 = 𝐴
 
Theoremissetri 3087* A way to say "𝐴 is a set" (inference rule). (Contributed by NM, 21-Jun-1993.)
𝑥 𝑥 = 𝐴       𝐴 ∈ V
 
Theoremeqvisset 3088 A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 3084 and issetri 3087. (Contributed by BJ, 27-Apr-2019.)
(𝑥 = 𝐴𝐴 ∈ V)
 
Theoremelex 3089 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.)
(𝐴𝐵𝐴 ∈ V)
 
Theoremelexi 3090 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
𝐴𝐵       𝐴 ∈ V
 
Theoremelexd 3091 If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)       (𝜑𝐴 ∈ V)
 
Theoremelisset 3092* An element of a class exists. (Contributed by NM, 1-May-1995.)
(𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 
Theoremelex2 3093* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)
 
Theoremelex22 3094* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
 
Theoremprcnel 3095 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 → ¬ 𝐴𝑉)
 
Theoremralv 3096 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑)
 
Theoremrexv 3097 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
 
Theoremreuv 3098 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
(∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑)
 
Theoremrmov 3099 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*𝑥 ∈ V 𝜑 ↔ ∃*𝑥𝜑)
 
Theoremrabab 3100 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} = {𝑥𝜑}
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