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Type | Label | Description |
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Statement | ||
Theorem | vtoclALT 3101* | Alternate proof of vtocl 3100. Shorter but requires more axioms. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | vtocl2 3102* | Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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Theorem | vtocl3 3103* | Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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Theorem | vtoclb 3104* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.) |
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Theorem | vtoclgf 3105 | Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.) |
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Theorem | vtoclg1f 3106* | Version of vtoclgf 3105 with one non-freeness hypothesis replaced with a dv condition, thus avoiding dependency on ax-11 1920 and ax-13 2091. (Contributed by BJ, 1-May-2019.) |
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Theorem | vtoclg 3107* | Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) |
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Theorem | vtoclbg 3108* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.) |
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Theorem | vtocl2gf 3109 | Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.) |
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Theorem | vtocl3gf 3110 | Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
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Theorem | vtocl2g 3111* | Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) |
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Theorem | vtoclgaf 3112* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.) |
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Theorem | vtoclga 3113* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.) |
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Theorem | vtocl2gaf 3114* | Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.) |
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Theorem | vtocl2ga 3115* | Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) |
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Theorem | vtocl3gaf 3116* | Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.) |
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Theorem | vtocl3ga 3117* | Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) |
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Theorem | vtocl4g 3118* | Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) |
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Theorem | vtocl4ga 3119* | Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) |
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Theorem | vtocleg 3120* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.) |
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Theorem | vtoclegft 3121* | Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3122.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) |
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Theorem | vtoclef 3122* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.) |
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Theorem | vtocle 3123* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) |
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Theorem | vtoclri 3124* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.) |
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Theorem | spcimgft 3125 | A closed version of spcimgf 3127. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | spcgft 3126 | A closed version of spcgf 3129. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.) |
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Theorem | spcimgf 3127 | Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | spcimegf 3128 | Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | spcgf 3129 | 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.) |
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Theorem | spcegf 3130 | Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.) |
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Theorem | spcimdv 3131* | Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | spcdv 3132* | Rule of specialization, using implicit substitution. Analogous to rspcdv 3153. (Contributed by David Moews, 1-May-2017.) |
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Theorem | spcimedv 3133* | Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | spcgv 3134* | Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) |
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Theorem | spcegv 3135* | Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.) |
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Theorem | spc2egv 3136* | Existential specialization with two quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.) |
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Theorem | spc2gv 3137* | Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.) |
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Theorem | spc3egv 3138* | Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) |
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Theorem | spc3gv 3139* | Specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) |
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Theorem | spcv 3140* | Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.) |
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Theorem | spcev 3141* | Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
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Theorem | spc2ev 3142* | Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.) |
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Theorem | rspct 3143* | A closed version of rspc 3144. (Contributed by Andrew Salmon, 6-Jun-2011.) |
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Theorem | rspc 3144* | Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) |
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Theorem | rspce 3145* | Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.) |
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Theorem | rspcv 3146* | Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) |
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Theorem | rspccv 3147* | Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.) |
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Theorem | rspcva 3148* | Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.) |
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Theorem | rspccva 3149* | Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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Theorem | rspcev 3150* | Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) |
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Theorem | rspcimdv 3151* | Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | rspcimedv 3152* | Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
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Theorem | rspcdv 3153* | Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.) |
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Theorem | rspcedv 3154* | Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.) |
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Theorem | rspcedvd 3155* | Restricted existential specialization, using implicit substitution. Variant of rspcdv 3153. (Contributed by AV, 27-Nov-2019.) |
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Theorem | rspcedeq1vd 3156* | Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3155 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.) |
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Theorem | rspcedeq2vd 3157* | Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3155 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.) |
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Theorem | rspc2 3158* | 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.) |
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Theorem | rspc2v 3159* | 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.) |
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Theorem | rspc2va 3160* | 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.) |
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Theorem | rspc2ev 3161* | 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.) |
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Theorem | rspc3v 3162* | 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.) |
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Theorem | rspc3ev 3163* | 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.) |
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Theorem | ralxpxfr2d 3164* | Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
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Theorem | rexraleqim 3165* | Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.) |
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Theorem | eqvinc 3166* | A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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Theorem | eqvincf 3167 | A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.) |
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Theorem | alexeqg 3168* |
Two ways to express substitution of ![]() ![]() ![]() |
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Theorem | ceqex 3169* | Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.) |
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Theorem | ceqsexg 3170* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.) |
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Theorem | ceqsexgv 3171* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) |
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Theorem | ceqsrexv 3172* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.) |
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Theorem | ceqsrexbv 3173* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) |
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Theorem | ceqsrex2v 3174* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.) |
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Theorem | clel2 3175* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
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Theorem | clel3g 3176* | An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.) |
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Theorem | clel3 3177* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
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Theorem | clel4 3178* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
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Theorem | pm13.183 3179* |
Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only ![]() |
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Theorem | rr19.3v 3180* | Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 3862 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.) |
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Theorem | rr19.28v 3181* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 3864 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.) |
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Theorem | elabgt 3182* | Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3186.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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Theorem | elabgf 3183 | 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.) |
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Theorem | elabf 3184* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | elab 3185* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) |
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Theorem | elabg 3186* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) |
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Theorem | elab2g 3187* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
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Theorem | elab2 3188* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
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Theorem | elab4g 3189* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |
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Theorem | elab3gf 3190 | Membership in a class abstraction, with a weaker antecedent than elabgf 3183. (Contributed by NM, 6-Sep-2011.) |
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Theorem | elab3g 3191* | Membership in a class abstraction, with a weaker antecedent than elabg 3186. (Contributed by NM, 29-Aug-2006.) |
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Theorem | elab3 3192* | Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |
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Theorem | elrabi 3193* | Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
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Theorem | elrabf 3194 | 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.) |
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Theorem | elrab3t 3195* | Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3197.) (Contributed by Thierry Arnoux, 31-Aug-2017.) |
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Theorem | elrab 3196* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) |
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Theorem | elrab3 3197* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.) |
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Theorem | elrab2 3198* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |
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Theorem | ralab 3199* | Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | ralrab 3200* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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