Home Metamath Proof ExplorerTheorem List (p. 32 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27759) Hilbert Space Explorer (27760-29284) Users' Mathboxes (29285-42322)

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

Theoremrexrot4 3101* Rotate four restricted existential quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷 𝜑 ↔ ∃𝑧𝐶𝑤𝐷𝑥𝐴𝑦𝐵 𝜑)

Theoremralcom2 3102* Commutation of restricted universal quantifiers. Note that 𝑥 and 𝑦 need not be disjoint (this makes the proof longer). If 𝑥 and 𝑦 are disjoint, then one may use ralcom 3096. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
(∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑦𝐴𝑥𝐴 𝜑)

Theoremralcom3 3103 A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))

Theoremreean 3104* Rearrange restricted existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))

Theoremreeanv 3105* Rearrange restricted existential quantifiers. (Contributed by NM, 9-May-1999.)
(∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))

Theorem3reeanv 3106* Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝜑𝜓𝜒) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓 ∧ ∃𝑧𝐶 𝜒))

Theorem2ralor 3107* Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))

Theoremnfreu1 3108 The setvar 𝑥 is not free in ∃!𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.)
𝑥∃!𝑥𝐴 𝜑

Theoremnfrmo1 3109 The setvar 𝑥 is not free in ∃*𝑥𝐴𝜑. (Contributed by NM, 16-Jun-2017.)
𝑥∃*𝑥𝐴 𝜑

Theoremnfreud 3110 Deduction version of nfreu 3112. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝐴 𝜓)

Theoremnfrmod 3111 Deduction version of nfrmo 3113. (Contributed by NM, 17-Jun-2017.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝐴 𝜓)

Theoremnfreu 3112 Bound-variable hypothesis builder for restricted unique existence. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.)
𝑥𝐴    &   𝑥𝜑       𝑥∃!𝑦𝐴 𝜑

Theoremnfrmo 3113 Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)
𝑥𝐴    &   𝑥𝜑       𝑥∃*𝑦𝐴 𝜑

Theoremrabid 3114 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 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))

Theoremrabidim1 3115 Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)

Theoremrabid2 3116* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)

Theoremrabid2f 3117 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.)
𝑥𝐴       (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)

Theoremrabbi 3118 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 3186. (Contributed by NM, 25-Nov-2013.)
(∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabswap 3119 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
{𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}

Theoremnfrab1 3120 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
𝑥{𝑥𝐴𝜑}

Theoremnfrab 3121 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 3122 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 3123* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))

Theoremreubidv 3124* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))

Theoremreubiia 3125 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
(𝑥𝐴 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)

Theoremreubii 3126 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
(𝜑𝜓)       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐴 𝜓)

Theoremrmobida 3127 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))

Theoremrmobidva 3128* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))

Theoremrmobidv 3129* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐴 𝜒))

Theoremrmobiia 3130 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
(𝑥𝐴 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)

Theoremrmobii 3131 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
(𝜑𝜓)       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐴 𝜓)

Theoremraleqf 3132 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 3133 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 3134 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 3135 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 3136* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))

Theoremrexeq 3137* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
(𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))

Theoremreueq1 3138* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))

Theoremrmoeq1 3139* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))

Theoremraleqi 3140* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐴 = 𝐵       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑)

Theoremrexeqi 3141* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐴 = 𝐵       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑)

Theoremraleqdv 3142* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))

Theoremrexeqdv 3143* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜓))

Theoremraleqbi1dv 3144* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))

Theoremrexeqbi1dv 3145* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓))

Theoremreueqd 3146* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))

Theoremrmoeqd 3147* 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 3148 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 3149 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 3150* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremrexeqbidv 3151* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremraleqbidva 3152* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremrexeqbidva 3153* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremraleleq 3154* 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 3155* Alternate proof of raleleq 3154 using ralel 2922, 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 3156 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝜑 → ∃*𝑥𝐴 𝜑)

Theoremreu5 3157 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 3158 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
(∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)

Theoremreurmo 3159 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
(∃!𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)

Theoremrmo5 3160 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))

Theoremnrexrmo 3161 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
(¬ ∃𝑥𝐴 𝜑 → ∃*𝑥𝐴 𝜑)

Theoremreueubd 3162* Restricted existential uniqueness is equivalent with existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021.)
((𝜑𝜓) → 𝑥𝑉)       (𝜑 → (∃!𝑥𝑉 𝜓 ↔ ∃!𝑥𝜓))

Theoremcbvralf 3163 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 3164 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 3165* 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 3166* 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 3167* 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 3168* 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 3169* 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 3170* 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 3171* 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 3172* 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 3173* 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 3174* 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 3175* 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 3176* 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 3177* 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 3178* 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 3179* 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 3180* 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 3181* 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 3182* 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 3183 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑥𝐴𝜓}

Theoremrabbidva2 3184* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabbia2 3185 Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒))       {𝑥𝐴𝜓} = {𝑥𝐵𝜒}

Theoremrabbidva 3186* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabbidv 3187* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrabeqf 3188 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})

Theoremrabeqif 3189 Equality theorem for restricted class abstractions. Inference form of rabeqf 3188. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴    &   𝑥𝐵    &   𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Theoremrabeq 3190* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
(𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})

Theoremrabeqi 3191* Equality theorem for restricted class abstractions. Inference form of rabeq 3190. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝐵       {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Theoremrabeqdv 3192* Equality of restricted class abstractions. Deduction form of rabeq 3190. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})

Theoremrabeqbidv 3193* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabeqbidva 3194* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Theoremrabeq2i 3195 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴 ↔ (𝑥𝐵𝜑))

Theoremcbvrab 3196 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 3197* 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 3198 Extend class notation to include the universal class symbol.
class V

Theoremvjust 3199 Soundness justification theorem for df-v 3200. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}

Definitiondf-v 3200 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 4794 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 3205 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}

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
 Copyright terms: Public domain < Previous  Next >