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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbcor 2801 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
 [.  ].  [.  ].  [.  ].
 
Theoremsbcorg 2802 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
 V  [.  ].  [.  ].  [.  ].
 
Theoremsbcbig 2803 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
 V  [.  ].  [.  ]. 
 [.  ].
 
Theoremsbcal 2804* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
 [.  ].  [.  ].
 
Theoremsbcalg 2805* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 V  [.  ].  [.  ].
 
Theoremsbcex2 2806* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
 [.  ].  [.  ].
 
Theoremsbcexg 2807* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
 V  [.  ].  [.  ].
 
Theoremsbceqal 2808* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
 V
 
Theoremsbeqalb 2809* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
 V
 
Theoremsbcbid 2810 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

 F/   &       =>     [.  ]. 
 [.  ].
 
Theoremsbcbidv 2811* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
   =>     [.  ]. 
 [.  ].
 
Theoremsbcbii 2812 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
   =>     [.  ].  [.  ].
 
Theoremeqsbc3r 2813* eqsbc3 2796 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)
 [.  ]. C  C
 
Theoremsbc3ang 2814 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 V  [.  ].  [.  ].  [.  ].  [.  ].
 
Theoremsbcel1gv 2815* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 V  [.  ].
 
Theoremsbcel2gv 2816* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 V  [.  ].
 
Theoremsbcimdv 2817* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
   =>     V  [.  ].  [.  ].
 
Theoremsbctt 2818 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
 V  F/  [.  ].
 
Theoremsbcgf 2819 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/   =>     V  [.  ].
 
Theoremsbc19.21g 2820 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

 F/   =>     V  [.  ].  [.  ].
 
Theoremsbcg 2821* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2819. (Contributed by Alan Sare, 10-Nov-2012.)
 V  [.  ].
 
Theoremsbc2iegf 2822* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)

 F/   &     F/   &     F/  W   &       =>     V  W  [.  ]. [.  ].
 
Theoremsbc2ie 2823* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
 _V   &     _V   &       =>     [.  ]. [.  ].
 
Theoremsbc2iedv 2824* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 _V   &     _V   &       =>     [.  ]. [.  ].
 
Theoremsbc3ie 2825* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
 _V   &     _V   &     C  _V   &     C    =>     [.  ]. [.  ]. [. C  ].
 
Theoremsbccomlem 2826* Lemma for sbccom 2827. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
 [.  ]. [.  ].  [.  ]. [.  ].
 
Theoremsbccom 2827* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 [.  ]. [.  ].  [.  ]. [.  ].
 
Theoremsbcralt 2828* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
 V  F/_  [.  ].  [.  ].
 
Theoremsbcrext 2829* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 V  F/_  [.  ].  [.  ].
 
Theoremsbcralg 2830* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 V  [.  ].  [.  ].
 
Theoremsbcrexg 2831* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 V  [.  ].  [.  ].
 
Theoremsbcreug 2832* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
 V  [.  ].  [.  ].
 
Theoremsbcabel 2833* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
 F/_   =>     V  [.  ]. {  | 
 }  {  |  [.  ]. }
 
Theoremrspsbc 2834* 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 1655 and spsbc 2769. See also rspsbca 2835 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 [.  ].
 
Theoremrspsbca 2835* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
 [.  ].
 
Theoremrspesbca 2836* Existence form of rspsbca 2835. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 [.  ].
 
Theoremspesbc 2837 Existence form of spsbc 2769. (Contributed by Mario Carneiro, 18-Nov-2016.)
 [.  ].
 
Theoremspesbcd 2838 form of spsbc 2769. (Contributed by Mario Carneiro, 9-Feb-2017.)
 [.  ].   =>   
 
Theoremsbcth2 2839* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
   =>     [.  ].
 
Theoremra5 2840 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 1473. (Contributed by NM, 16-Jan-2004.)

 F/   =>   
 
Theoremrmo2ilem 2841* Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)

 F/   =>   
 
Theoremrmo2i 2842* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)

 F/   =>   
 
Theoremrmo3 2843* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

 F/   =>   
 
Theoremrmob 2844* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
   &     C    =>     C  C
 
Theoremrmoi 2845* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
   &     C    =>     C  C
 
2.1.10  Proper substitution of classes for sets into classes
 
Syntaxcsb 2846 Extend class notation to include the proper substitution of a class for a set into another class.
 [_  ]_
 
Definitiondf-csb 2847* 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 2758, to prevent ambiguity. Theorem sbcel1g 2863 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2872 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
 [_  ]_  {  |  [.  ].  }
 
Theoremcsb2 2848* 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 2849 Analog of dfsbcq 2760 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 [_  ]_ C  [_  ]_ C
 
Theoremcbvcsb 2850 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/_ C   &     F/_ D   &     C  D   =>     [_  ]_ C  [_  ]_ D
 
Theoremcbvcsbv 2851* 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.)
 C   =>     [_  ]_  [_  ]_ C
 
Theoremcsbeq1d 2852 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
   =>     [_  ]_ C  [_  ]_ C
 
Theoremcsbid 2853 Analog of sbid 1654 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 [_  ]_
 
Theoremcsbeq1a 2854 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 [_  ]_
 
Theoremcsbco 2855* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
 [_  ]_ [_  ]_  [_  ]_
 
Theoremcsbtt 2856 Substitution doesn't affect a constant (in which is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
 V  F/_  [_  ]_
 
Theoremcsbconstgf 2857 Substitution doesn't affect a constant (in which is not free). (Contributed by NM, 10-Nov-2005.)
 F/_   =>     V  [_  ]_
 
Theoremcsbconstg 2858* Substitution doesn't affect a constant (in which is not free). csbconstgf 2857 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
 V  [_  ]_
 
Theoremsbcel12g 2859 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 V  [.  ].  C  [_  ]_  [_  ]_ C
 
Theoremsbceqg 2860 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 V  [.  ].  C  [_  ]_  [_  ]_ C
 
Theoremsbcnel12g 2861 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
 V  [.  ].  e/  C  [_  ]_  e/  [_  ]_ C
 
Theoremsbcne12g 2862 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
 V  [.  ].  =/=  C  [_  ]_  =/=  [_  ]_ C
 
Theoremsbcel1g 2863* Move proper substitution in and out of a membership relation. Note that the scope of  [.  ]. is the wff  C, whereas the scope of  [_  ]_ is the class . (Contributed by NM, 10-Nov-2005.)
 V  [.  ].  C  [_  ]_  C
 
Theoremsbceq1g 2864* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
 V  [.  ].  C  [_  ]_  C
 
Theoremsbcel2g 2865* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
 V  [.  ].  C  [_  ]_ C
 
Theoremsbceq2g 2866* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
 V  [.  ].  C  [_  ]_ C
 
Theoremcsbcomg 2867* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
 V  W  [_  ]_ [_  ]_ C  [_  ]_ [_  ]_ C
 
Theoremcsbeq2d 2868 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

 F/   &     C   =>     [_  ]_  [_  ]_ C
 
Theoremcsbeq2dv 2869* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 C   =>     [_  ]_  [_  ]_ C
 
Theoremcsbeq2i 2870 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 C   =>     [_  ]_  [_  ]_ C
 
Theoremcsbvarg 2871 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
 V  [_  ]_
 
Theoremsbccsbg 2872* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
 V  [.  ].  [_  ]_ {  |  }
 
Theoremsbccsb2g 2873 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
 V  [.  ].  [_  ]_ {  |  }
 
Theoremnfcsb1d 2874 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 F/_   =>     F/_ [_  ]_
 
Theoremnfcsb1 2875 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 F/_   =>     F/_ [_  ]_
 
Theoremnfcsb1v 2876* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 F/_ [_  ]_
 
Theoremnfcsbd 2877 Deduction version of nfcsb 2878. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

 F/   &     F/_   &     F/_   =>     F/_ [_  ]_
 
Theoremnfcsb 2878 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 F/_   &     F/_   =>     F/_ [_  ]_
 
Theoremcsbhypf 2879* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2597 for class substitution version. (Contributed by NM, 19-Dec-2008.)
 F/_   &     F/_ C   &     C   =>     [_  ]_  C
 
Theoremcsbiebt 2880* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2884.) (Contributed by NM, 11-Nov-2005.)
 V  F/_ C  C  [_  ]_  C
 
Theoremcsbiedf 2881* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

 F/   &     F/_ C   &     V   &     C   =>     [_  ]_  C
 
Theoremcsbieb 2882* Bidirectional conversion between an implicit class substitution hypothesis  C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
 _V   &     F/_ C   =>     C  [_  ]_  C
 
Theoremcsbiebg 2883* Bidirectional conversion between an implicit class substitution hypothesis  C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 F/_ C   =>     V  C  [_  ]_  C
 
Theoremcsbiegf 2884* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 V  F/_ C   &     C   =>     V  [_  ]_  C
 
Theoremcsbief 2885* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 _V   &     F/_ C   &     C   =>     [_  ]_  C
 
Theoremcsbied 2886* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
 V   &     C   =>     [_  ]_  C
 
Theoremcsbied2 2887* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
 V   &       &     C  D   =>     [_  ]_ C  D
 
Theoremcsbie2t 2888* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2889). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
 _V   &     _V   =>     C  D  [_  ]_ [_  ]_ C  D
 
Theoremcsbie2 2889* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
 _V   &     _V   &     C  D   =>     [_  ]_ [_  ]_ C  D
 
Theoremcsbie2g 2890* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2791 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
 C   &     C  D   =>     V  [_  ]_  D
 
Theoremsbcnestgf 2891 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
 V  F/  [.  ].
 [.  ]. 
 [. [_  ]_  ].
 
Theoremcsbnestgf 2892 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
 V  F/_ C  [_  ]_ [_  ]_ C  [_
 [_  ]_  ]_ C
 
Theoremsbcnestg 2893* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 V  [.  ]. [.  ].  [. [_  ]_  ].
 
Theoremcsbnestg 2894* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
 V  [_  ]_
 [_  ]_ C  [_ [_  ]_  ]_ C
 
Theoremcsbnest1g 2895 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 V  [_  ]_
 [_  ]_ C  [_ [_  ]_  ]_ C
 
Theoremcsbidmg 2896* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
 V  [_  ]_
 [_  ]_  [_  ]_
 
Theoremsbcco3g 2897* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
 C   =>     V  [.  ]. [.  ].  [. C  ].
 
Theoremcsbco3g 2898* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
 C   =>     V  [_  ]_
 [_  ]_ D  [_ C  ]_ D
 
Theoremrspcsbela 2899* Special case related to rspsbc 2834. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
 C  D  [_  ]_ C  D
 
Theoremsbnfc2 2900* Two ways of expressing " is (effectively) not free in ." (Contributed by Mario Carneiro, 14-Oct-2016.)
 F/_  [_  ]_  [_  ]_
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