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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelrabsf 2801 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2696 has implicit substitution). The hypothesis specifies that must not be a free variable in . (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremeqsbc3 2802* Substitution applied to an atomic wff. Set theory version of eqsb3 2141. (Contributed by Andrew Salmon, 29-Jun-2011.)

Theoremsbcng 2803 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Theoremsbcimg 2804 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)

Theoremsbcan 2805 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)

Theoremsbcang 2806 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)

Theoremsbcor 2807 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)

Theoremsbcorg 2808 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)

Theoremsbcbig 2809 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremsbcal 2810* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)

Theoremsbcalg 2811* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Theoremsbcex2 2812* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Theoremsbcexg 2813* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Theoremsbceqal 2814* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)

Theoremsbeqalb 2815* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)

Theoremsbcbid 2816 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Theoremsbcbidv 2817* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Theoremsbcbii 2818 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)

Theoremeqsbc3r 2819* eqsbc3 2802 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremsbc3ang 2820 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcel1gv 2821* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcel2gv 2822* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcimdv 2823* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)

Theoremsbctt 2824 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremsbcgf 2825 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.)

Theoremsbc19.21g 2826 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Theoremsbcg 2827* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2825. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremsbc2iegf 2828* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2ie 2829* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2iedv 2830* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbc3ie 2831* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremsbccomlem 2832* Lemma for sbccom 2833. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)

Theoremsbccom 2833* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbcralt 2834* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)

Theoremsbcrext 2835* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremsbcralg 2836* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcrex 2837* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Theoremsbcreug 2838* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)

Theoremsbcabel 2839* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)

Theoremrspsbc 2840* 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 1658 and spsbc 2775. See also rspsbca 2841 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremrspsbca 2841* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)

Theoremrspesbca 2842* Existence form of rspsbca 2841. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremspesbc 2843 Existence form of spsbc 2775. (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremspesbcd 2844 form of spsbc 2775. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbcth2 2845* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremra5 2846 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 1476. (Contributed by NM, 16-Jan-2004.)

Theoremrmo2ilem 2847* Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)

Theoremrmo2i 2848* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)

Theoremrmo3 2849* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Theoremrmob 2850* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)

Theoremrmoi 2851* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

2.1.10  Proper substitution of classes for sets into classes

Syntaxcsb 2852 Extend class notation to include the proper substitution of a class for a set into another class.

Definitiondf-csb 2853* 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 2764, to prevent ambiguity. Theorem sbcel1g 2869 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2878 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)

Theoremcsb2 2854* 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 2855 Analog of dfsbcq 2766 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcbvcsb 2856 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.)

Theoremcbvcsbv 2857* 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 2858 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)

Theoremcsbid 2859 Analog of sbid 1657 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbeq1a 2860 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbco 2861* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbtt 2862 Substitution doesn't affect a constant (in which is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremcsbconstgf 2863 Substitution doesn't affect a constant (in which is not free). (Contributed by NM, 10-Nov-2005.)

Theoremcsbconstg 2864* Substitution doesn't affect a constant (in which is not free). csbconstgf 2863 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremsbcel12g 2865 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbceqg 2866 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcnel12g 2867 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)

Theoremsbcne12g 2868 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)

Theoremsbcel1g 2869* 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 2870* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremsbcel2g 2871* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)

Theoremsbceq2g 2872* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremcsbcomg 2873* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)

Theoremcsbeq2d 2874 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2dv 2875* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2i 2876 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbvarg 2877 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)

Theoremsbccsbg 2878* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Theoremsbccsb2g 2879 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Theoremnfcsb1d 2880 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1 2881 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1v 2882* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsbd 2883 Deduction version of nfcsb 2884. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb 2884 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremcsbhypf 2885* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2603 for class substitution version. (Contributed by NM, 19-Dec-2008.)

Theoremcsbiebt 2886* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2890.) (Contributed by NM, 11-Nov-2005.)

Theoremcsbiedf 2887* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Theoremcsbieb 2888* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)

Theoremcsbiebg 2889* 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.)

Theoremcsbiegf 2890* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbief 2891* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbied 2892* 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 2893* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcsbie2t 2894* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2895). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbie2 2895* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Theoremcsbie2g 2896* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2797 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremsbcnestgf 2897 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestgf 2898 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Theoremsbcnestg 2899* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestg 2900* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

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