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

Theoremsbcel21v 3001* Class substitution into a membership relation. One direction of sbcel2gv 3000 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Theoremsbcimdv 3002* Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1437). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)

Theoremsbctt 3003 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremsbcgf 3004 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 3005 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Theoremsbcg 3006* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3004. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremsbc2iegf 3007* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2ie 3008* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2iedv 3009* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbc3ie 3010* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremsbccomlem 3011* Lemma for sbccom 3012. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)

Theoremsbccom 3012* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbcralt 3013* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)

Theoremsbcrext 3014* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremsbcralg 3015* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcrex 3016* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Theoremsbcreug 3017* Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.)

Theoremsbcabel 3018* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)

Theoremrspsbc 3019* 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 1755 and spsbc 2948. See also rspsbca 3020 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremrspsbca 3020* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)

Theoremrspesbca 3021* Existence form of rspsbca 3020. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremspesbc 3022 Existence form of spsbc 2948. (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremspesbcd 3023 form of spsbc 2948. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbcth2 3024* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremra5 3025 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 1564. (Contributed by NM, 16-Jan-2004.)

Theoremrmo2ilem 3026* Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.)

Theoremrmo2i 3027* Condition implying restricted at-most-one quantifier. (Contributed by NM, 17-Jun-2017.)

Theoremrmo3 3028* Restricted at-most-one quantifier using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Theoremrmob 3029* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)

Theoremrmoi 3030* 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 3031 Extend class notation to include the proper substitution of a class for a set into another class.

Definitiondf-csb 3032* 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 2937, to prevent ambiguity. Theorem sbcel1g 3050 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3060 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)

Theoremcsb2 3033* 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 3034 Analog of dfsbcq 2939 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcbvcsbw 3035* Version of cbvcsb 3036 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)

Theoremcbvcsb 3036 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 3037* 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 3038 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)

Theoremcsbid 3039 Analog of sbid 1754 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbeq1a 3040 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbco 3041* Composition law for chained substitutions into a class.

Use the weaker csbcow 3042 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)

Theoremcsbcow 3042* Composition law for chained substitutions into a class. Version of csbco 3041 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 25-Aug-2024.)

Theoremcsbtt 3043 Substitution doesn't affect a constant (in which is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremcsbconstgf 3044 Substitution doesn't affect a constant (in which is not free). (Contributed by NM, 10-Nov-2005.)

Theoremcsbconstg 3045* Substitution doesn't affect a constant (in which is not free). csbconstgf 3044 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremsbcel12g 3046 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbceqg 3047 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcnel12g 3048 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)

Theoremsbcne12g 3049 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)

Theoremsbcel1g 3050* 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 3051* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremsbcel2g 3052* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)

Theoremsbceq2g 3053* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremcsbcomg 3054* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)

Theoremcsbeq2 3055 Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Theoremcsbeq2d 3056 Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2dv 3057* Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2i 3058 Formula-building inference for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbvarg 3059 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)

Theoremsbccsbg 3060* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Theoremsbccsb2g 3061 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Theoremnfcsb1d 3062 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1 3063 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1v 3064* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbcdw 3065* Version of nfsbcd 2956 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)

Theoremnfcsbd 3066 Deduction version of nfcsb 3068. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsbw 3067* Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3068 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)

Theoremnfcsb 3068 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremcsbhypf 3069* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2761 for class substitution version. (Contributed by NM, 19-Dec-2008.)

Theoremcsbiebt 3070* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3074.) (Contributed by NM, 11-Nov-2005.)

Theoremcsbiedf 3071* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Theoremcsbieb 3072* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)

Theoremcsbiebg 3073* 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 3074* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbief 3075* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbie 3076* Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)

Theoremcsbied 3077* 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 3078* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcsbie2t 3079* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3080). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbie2 3080* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Theoremcsbie2g 3081* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2971 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremsbcnestgf 3082 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestgf 3083 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Theoremsbcnestg 3084* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestg 3085* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Theoremcsbnest1g 3086 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbidmg 3087* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)

Theoremsbcco3g 3088* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Theoremcsbco3g 3089* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Theoremrspcsbela 3090* Special case related to rspsbc 3019. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Theoremsbnfc2 3091* Two ways of expressing " is (effectively) not free in ." (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremcsbabg 3092* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremcbvralcsf 3093 A more general version of cbvralf 2674 that doesn't require and to be distinct from or . Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvrexcsf 3094 A more general version of cbvrexf 2675 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.)

Theoremcbvreucsf 3095 A more general version of cbvreuv 2682 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvrabcsf 3096 A more general version of cbvrab 2710 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvralv2 3097* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

Theoremcbvrexv2 3098* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

2.1.11  Define basic set operations and relations

Syntaxcdif 3099 Extend class notation to include class difference (read: " minus ").

Syntaxcun 3100 Extend class notation to include union of two classes (read: " union ").

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