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

Theoremvtocl2ga 3101* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl3gaf 3102* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtocl3ga 3103* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl4g 3104* Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)

Theoremvtocl4ga 3105* Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)

Theoremvtocleg 3106* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.)

Theoremvtoclegft 3107* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3108.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtoclef 3108* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)

Theoremvtocle 3109* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)

Theoremvtoclri 3110* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)

Theoremspcimgft 3111 A closed version of spcimgf 3113. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgft 3112 A closed version of spcgf 3115. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremspcimgf 3113 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcimegf 3114 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgf 3115 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)

Theoremspcegf 3116 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)

Theoremspcimdv 3117* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcdv 3118* Rule of specialization, using implicit substitution. Analogous to rspcdv 3139. (Contributed by David Moews, 1-May-2017.)

Theoremspcimedv 3119* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgv 3120* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

Theoremspcegv 3121* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)

Theoremspc2egv 3122* Existential specialization with two quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremspc2gv 3123* Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Theoremspc3egv 3124* Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremspc3gv 3125* Specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremspcv 3126* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)

Theoremspcev 3127* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremspc2ev 3128* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremrspct 3129* A closed version of rspc 3130. (Contributed by Andrew Salmon, 6-Jun-2011.)

Theoremrspc 3130* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrspce 3131* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrspcv 3132* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrspccv 3133* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)

Theoremrspcva 3134* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)

Theoremrspccva 3135* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrspcev 3136* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrspcimdv 3137* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrspcimedv 3138* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrspcdv 3139* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremrspcedv 3140* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremrspcda 3141* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.)

Theoremrspcdva 3142* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Theoremrspcedvd 3143* Restricted existential specialization, using implicit substitution. Variant of rspcdv 3139. (Contributed by AV, 27-Nov-2019.)

Theoremrspcedeq1vd 3144* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3143 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)

Theoremrspcedeq2vd 3145* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3143 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)

Theoremrspc2 3146* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)

Theoremrspc2v 3147* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)

Theoremrspc2va 3148* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)

Theoremrspc2ev 3149* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)

Theoremrspc3v 3150* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)

Theoremrspc3ev 3151* 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)

Theoremralxpxfr2d 3152* Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Theoremrexraleqim 3153* Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)

Theoremeqvinc 3154* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremeqvincf 3155 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)

Theoremalexeqg 3156* Two ways to express substitution of for in . This is the analogue for classes of sb56 2096. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.)

Theoremceqex 3157* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.)

Theoremceqsexg 3158* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)

Theoremceqsexgv 3159* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)

Theoremceqsrexv 3160* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)

Theoremceqsrexbv 3161* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)

Theoremceqsrex2v 3162* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)

Theoremclel2 3163* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theoremclel3g 3164* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)

Theoremclel3 3165* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theoremclel4 3166* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theorempm13.183 3167* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremrr19.3v 3168* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 3853 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)

Theoremrr19.28v 3169* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 3855 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)

Theoremelabgt 3170* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3174.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremelabgf 3171 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremelabf 3172* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremelab 3173* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)

Theoremelabg 3174* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)

Theoremelab2g 3175* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Theoremelab2 3176* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Theoremelab4g 3177* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)

Theoremelab3gf 3178 Membership in a class abstraction, with a weaker antecedent than elabgf 3171. (Contributed by NM, 6-Sep-2011.)

Theoremelab3g 3179* Membership in a class abstraction, with a weaker antecedent than elabg 3174. (Contributed by NM, 29-Aug-2006.)

Theoremelab3 3180* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)

Theoremelrabi 3181* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Theoremelrabf 3182 Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)

Theoremelrab3t 3183* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3185.) (Contributed by Thierry Arnoux, 31-Aug-2017.)

Theoremelrab 3184* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)

Theoremelrab3 3185* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)

Theoremelrab2 3186* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)

Theoremralab 3187* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremralrab 3188* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremrexab 3189* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremrexrab 3190* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremralab2 3191* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremralrab2 3192* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremrexab2 3193* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremrexrab2 3194* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremabidnf 3195* Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)

Theoremdedhb 3196* A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1999 and nfab 2616 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 3195 is useful. (Contributed by NM, 8-Dec-2006.)

Theoremeqeu 3197* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)

Theoremeueq 3198* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)

Theoremeueq1 3199* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremeueq2 3200* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)

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