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Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgencbvex2 3001* Restatement of gencbvex 3000 with weaker hypotheses. (Contributed by Jeffrey Hankins, 6-Dec-2006.)

Theoremgencbval 3002* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)

Theoremsbhypf 3003* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3288. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremvtoclgft 3004 Closed theorem form of vtoclgf 3012. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremvtocldf 3005 Implicit substitution of a class for a set variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremvtocld 3006* Implicit substitution of a class for a set variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremvtoclf 3007* Implicit substitution of a class for a set variable. This is a generalization of chvar 1969. (Contributed by NM, 30-Aug-1993.)

Theoremvtocl 3008* Implicit substitution of a class for a set variable. (Contributed by NM, 30-Aug-1993.)

Theoremvtocl2 3009* Implicit substitution of classes for set variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremvtocl3 3010* Implicit substitution of classes for set variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremvtoclb 3011* Implicit substitution of a class for a set variable. (Contributed by NM, 23-Dec-1993.)

Theoremvtoclgf 3012 Implicit substitution of a class for a set variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)

Theoremvtoclg 3013* Implicit substitution of a class expression for a set variable. (Contributed by NM, 17-Apr-1995.)

Theoremvtoclbg 3014* Implicit substitution of a class for a set variable. (Contributed by NM, 29-Apr-1994.)

Theoremvtocl2gf 3015 Implicit substitution of a class for a set variable. (Contributed by NM, 25-Apr-1995.)

Theoremvtocl3gf 3016 Implicit substitution of a class for a set variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremvtocl2g 3017* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 25-Apr-1995.)

Theoremvtoclgaf 3018* Implicit substitution of a class for a set variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremvtoclga 3019* Implicit substitution of a class for a set variable. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl2gaf 3020* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 10-Aug-2013.)

Theoremvtocl2ga 3021* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl3gaf 3022* Implicit substitution of 3 classes for 3 set variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtocl3ga 3023* Implicit substitution of 3 classes for 3 set variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocleg 3024* Implicit substitution of a class for a set variable. (Contributed by NM, 10-Jan-2004.)

Theoremvtoclegft 3025* Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 3026.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtoclef 3026* Implicit substitution of a class for a set variable. (Contributed by NM, 18-Aug-1993.)

Theoremvtocle 3027* Implicit substitution of a class for a set variable. (Contributed by NM, 9-Sep-1993.)

Theoremvtoclri 3028* Implicit substitution of a class for a set variable. (Contributed by NM, 21-Nov-1994.)

Theoremspcimgft 3029 A closed version of spcimgf 3031. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgft 3030 A closed version of spcgf 3033. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremspcimgf 3031 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcimegf 3032 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgf 3033 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 3034 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)

Theoremspcimdv 3035* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcdv 3036* Rule of specialization, using implicit substitution. Analogous to rspcdv 3057. (Contributed by David Moews, 1-May-2017.)

Theoremspcimedv 3037* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgv 3038* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

Theoremspcegv 3039* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)

Theoremspc2egv 3040* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremspc2gv 3041* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Theoremspc3egv 3042* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremspc3gv 3043* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremspcv 3044* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)

Theoremspcev 3045* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremspc2ev 3046* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremrspct 3047* A closed version of rspc 3048. (Contributed by Andrew Salmon, 6-Jun-2011.)

Theoremrspc 3048* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrspce 3049* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrspcv 3050* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrspccv 3051* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)

Theoremrspcva 3052* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)

Theoremrspccva 3053* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrspcev 3054* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrspcimdv 3055* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrspcimedv 3056* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrspcdv 3057* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremrspcedv 3058* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremrspc2 3059* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)

Theoremrspc2v 3060* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)

Theoremrspc2va 3061* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)

Theoremrspc2ev 3062* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)

Theoremrspc3v 3063* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)

Theoremrspc3ev 3064* 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)

Theoremeqvinc 3065* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremeqvincf 3066 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)

Theoremalexeq 3067* Two ways to express substitution of for in . (Contributed by NM, 2-Mar-1995.)

Theoremceqex 3068* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)

Theoremceqsexg 3069* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)

Theoremceqsexgv 3070* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)

Theoremceqsrexv 3071* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)

Theoremceqsrexbv 3072* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)

Theoremceqsrex2v 3073* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)

Theoremclel2 3074* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theoremclel3g 3075* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)

Theoremclel3 3076* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theoremclel4 3077* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theorempm13.183 3078* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremrr19.3v 3079* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3723 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)

Theoremrr19.28v 3080* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3725 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)

Theoremelabgt 3081* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3085.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremelabgf 3082 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 3083* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremelab 3084* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)

Theoremelabg 3085* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)

Theoremelab2g 3086* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Theoremelab2 3087* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Theoremelab4g 3088* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)

Theoremelab3gf 3089 Membership in a class abstraction, with a weaker antecedent than elabgf 3082. (Contributed by NM, 6-Sep-2011.)

Theoremelab3g 3090* Membership in a class abstraction, with a weaker antecedent than elabg 3085. (Contributed by NM, 29-Aug-2006.)

Theoremelab3 3091* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)

Theoremelrabi 3092* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Theoremelrabf 3093 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.)

Theoremelrab 3094* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)

Theoremelrab3 3095* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)

Theoremelrab2 3096* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)

Theoremralab 3097* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremralrab 3098* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremrexab 3099* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremrexrab 3100* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)

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