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Theorem List for New Foundations Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbcbii 3101 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
   =>     [.  ].  [.  ].
 
TheoremsbcbiiOLD 3102 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.)
   =>     [.  ].  [.  ].
 
Theoremeqsbc3r 3103* eqsbc3 3085 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD in set.mm using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
 [.  ].
 
Theoremsbc3ang 3104 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 [.  ].  [.  ].  [.  ].  [.  ].
 
Theoremsbcel1gv 3105* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 [.  ].
 
Theoremsbcel2gv 3106* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 [.  ].
 
Theoremsbcimdv 3107* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
   =>     [.  ].  [.  ].
 
Theoremsbctt 3108 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
 F/  [.  ].
 
Theoremsbcgf 3109 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

 F/   =>     [.  ].
 
Theoremsbc19.21g 3110 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

 F/   =>     [.  ].  [.  ].
 
Theoremsbcg 3111* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3109. (Contributed by Alan Sare, 10-Nov-2012.)
 [.  ].
 
Theoremsbc2iegf 3112* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)

 F/   &     F/   &     F/    &       =>     [.  ]. [.  ].
 
Theoremsbc2ie 3113* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
   &       &       =>     [.  ]. [.  ].
 
Theoremsbc2iedv 3114* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
   &       &       =>     [.  ]. [.  ].
 
Theoremsbc3ie 3115* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
   &       &       &       =>     [.  ]. [.  ]. [.  ].
 
Theoremsbccomlem 3116* Lemma for sbccom 3117. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
 [.  ]. [.  ].  [.  ]. [.  ].
 
Theoremsbccom 3117* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 [.  ]. [.  ].  [.  ]. [.  ].
 
Theoremsbcralt 3118* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
 F/_  [.  ].  [.  ].
 
Theoremsbcrext 3119* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 F/_  [.  ].  [.  ].
 
Theoremsbcralg 3120* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 [.  ].  [.  ].
 
Theoremsbcrexg 3121* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 [.  ].  [.  ].
 
Theoremsbcreug 3122* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
 [.  ].  [.  ].
 
Theoremsbcabel 3123* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
 F/_   =>     [.  ].  [.  ].
 
Theoremrspsbc 3124* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 2024 and spsbc 3058. See also rspsbca 3125 and rspcsbela 3195. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 [.  ].
 
Theoremrspsbca 3125* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
 [.  ].
 
Theoremrspesbca 3126* Existence form of rspsbca 3125. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 [.  ].
 
Theoremspesbc 3127 Existence form of spsbc 3058. (Contributed by Mario Carneiro, 18-Nov-2016.)
 [.  ].
 
Theoremspesbcd 3128 form of spsbc 3058. (Contributed by Mario Carneiro, 9-Feb-2017.)
 [.  ].   =>   
 
Theoremsbcth2 3129* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
   =>     [.  ].
 
Theoremra5 3130 Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1798. (Contributed by NM, 16-Jan-2004.)

 F/   =>   
 
Theoremrmo2 3131* Alternate definition of restricted "at most one." Note that is not equivalent to (in analogy to reu6 3025); to see this, let be the empty set. However, one direction of this pattern holds; see rmo2i 3132. (Contributed by NM, 17-Jun-2017.)

 F/   =>   
 
Theoremrmo2i 3132* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)

 F/   =>   
 
Theoremrmo3 3133* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

 F/   =>   
 
Theoremrmob 3134* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
   &       =>   
 
Theoremrmoi 3135* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
   &       =>   
 
2.1.9  Proper substitution of classes for sets into classes
 
Syntaxcsb 3136 Extend class notation to include the proper substitution of a class for a set into another class.
 
Definitiondf-csb 3137* Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3046, to prevent ambiguity. Theorem sbcel1g 3155 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3164 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
 [.  ].
 
Theoremcsb2 3138* Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that can be free in but cannot occur in . (Contributed by NM, 2-Dec-2013.)
 
Theoremcsbeq1 3139 Analog of dfsbcq 3048 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 
Theoremcbvcsb 3140 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on . (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
 F/_   &     F/_   &       =>   
 
Theoremcbvcsbv 3141* Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
   =>   
 
Theoremcsbeq1d 3142 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
   =>   
 
Theoremcsbid 3143 Analog of sbid 1922 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 
Theoremcsbeq1a 3144 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 
Theoremcsbco 3145* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
 
Theoremcsbexg 3146 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 
Theoremcsbex 3147 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
   &       =>   
 
Theoremcsbtt 3148 Substitution doesn't affect a constant (in which is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
 F/_
 
Theoremcsbconstgf 3149 Substitution doesn't affect a constant (in which is not free). (Contributed by NM, 10-Nov-2005.)
 F/_   =>   
 
Theoremcsbconstg 3150* Substitution doesn't affect a constant (in which is not free). csbconstgf 3149 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
 
Theoremsbcel12g 3151 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 [.  ].
 
Theoremsbceqg 3152 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 [.  ].
 
Theoremsbcnel12g 3153 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
 [.  ].
 
Theoremsbcne12g 3154 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
 [.  ].
 
Theoremsbcel1g 3155* Move proper substitution in and out of a membership relation. Note that the scope of  [.  ]. is the wff , whereas the scope of is the class . (Contributed by NM, 10-Nov-2005.)
 [.  ].
 
Theoremsbceq1g 3156* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
 [.  ].
 
Theoremsbcel2g 3157* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
 [.  ].
 
Theoremsbceq2g 3158* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
 [.  ].
 
Theoremcsbcomg 3159* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
 
Theoremcsbeq2d 3160 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

 F/   &       =>   
 
Theoremcsbeq2dv 3161* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
   =>   
 
Theoremcsbeq2i 3162 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
   =>   
 
Theoremcsbvarg 3163 The proper substitution of a class for set variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
 
Theoremsbccsbg 3164* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
 [.  ].
 
Theoremsbccsb2g 3165 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
 [.  ].
 
Theoremnfcsb1d 3166 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 F/_   =>     F/_
 
Theoremnfcsb1 3167 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 F/_   =>     F/_
 
Theoremnfcsb1v 3168* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 F/_
 
Theoremnfcsbd 3169 Deduction version of nfcsb 3170. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

 F/   &     F/_   &     F/_   =>     F/_
 
Theoremnfcsb 3170 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 F/_   &     F/_   =>     F/_
 
Theoremcsbhypf 3171* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2904 for class substitution version. (Contributed by NM, 19-Dec-2008.)
 F/_   &     F/_   &       =>   
 
Theoremcsbiebt 3172* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3176.) (Contributed by NM, 11-Nov-2005.)
 F/_
 
Theoremcsbiedf 3173* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

 F/   &     F/_   &       &       =>   
 
Theoremcsbieb 3174* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
   &     F/_   =>   
 
Theoremcsbiebg 3175* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 F/_   =>   
 
Theoremcsbiegf 3176* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 F/_   &       =>   
 
Theoremcsbief 3177* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
   &     F/_   &       =>   
 
Theoremcsbied 3178* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
   &       =>   
 
Theoremcsbied2 3179* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
   &       &       =>   
 
Theoremcsbie2t 3180* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3181). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
   &       =>   
 
Theoremcsbie2 3181* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
   &       &       =>   
 
Theoremcsbie2g 3182* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3080 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
   &       =>   
 
Theoremsbcnestgf 3183 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
 F/  [.  ].
 [.  ]. 
 [.  ].
 
Theoremcsbnestgf 3184 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
 F/_
 
Theoremsbcnestg 3185* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 [.  ]. [.  ].  [.  ].
 
Theoremcsbnestg 3186* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
 
TheoremcsbnestgOLD 3187* Nest the composition of two substitutions. (New usage is discouraged.) (Contributed by NM, 23-Nov-2005.)
 
Theoremcsbnest1g 3188 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 
Theoremcsbnest1gOLD 3189* Nest the composition of two substitutions. Obsolete as of 11-Nov-2016. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
 
Theoremcsbidmg 3190* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
 
Theoremsbcco3g 3191* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
   =>     [.  ]. [.  ].  [.  ].
 
Theoremsbcco3gOLD 3192* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.)
   =>     [.  ]. [.  ].  [.  ].
 
Theoremcsbco3g 3193* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
   =>   
 
Theoremcsbco3gOLD 3194* Composition of two class substitutions. Obsolete as of 11-Nov-2016. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.)
   =>   
 
Theoremrspcsbela 3195* Special case related to rspsbc 3124. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
 
Theoremsbnfc2 3196* Two ways of expressing " is (effectively) not free in ." (Contributed by Mario Carneiro, 14-Oct-2016.)
 F/_
 
Theoremcsbabg 3197* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 [.  ].
 
Theoremcbvralcsf 3198 A more general version of cbvralf 2829 that doesn't require and to be distinct from or . Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
 F/_   &     F/_   &     F/   &     F/   &       &       =>   
 
Theoremcbvrexcsf 3199 A more general version of cbvrexf 2830 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
 F/_   &     F/_   &     F/   &     F/   &       &       =>   
 
Theoremcbvreucsf 3200 A more general version of cbvreuv 2837 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
 F/_   &     F/_   &     F/   &     F/   &       &       =>   
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