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Type | Label | Description |
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Statement | ||
Theorem | elrabsf 3001 |
Membership in a restricted class abstraction, expressed with explicit
class substitution. (The variation elrabf 2891 has implicit substitution).
The hypothesis specifies that ![]() ![]() |
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Theorem | eqsbc1 3002* | Substitution for the left-hand side in an equality. Class version of eqsb1 2281. (Contributed by Andrew Salmon, 29-Jun-2011.) |
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Theorem | sbcng 3003 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
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Theorem | sbcimg 3004 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
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Theorem | sbcan 3005 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |
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Theorem | sbcang 3006 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |
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Theorem | sbcor 3007 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |
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Theorem | sbcorg 3008 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |
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Theorem | sbcbig 3009 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
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Theorem | sbcn1 3010 | Move negation in and out of class substitution. One direction of sbcng 3003 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
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Theorem | sbcim1 3011 | Distribution of class substitution over implication. One direction of sbcimg 3004 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
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Theorem | sbcbi1 3012 | Distribution of class substitution over biconditional. One direction of sbcbig 3009 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
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Theorem | sbcbi2 3013 | Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
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Theorem | sbcal 3014* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |
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Theorem | sbcalg 3015* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
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Theorem | sbcex2 3016* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
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Theorem | sbcexg 3017* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
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Theorem | sbceqal 3018* | A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.) |
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Theorem | sbeqalb 3019* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |
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Theorem | sbcbid 3020 | Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) |
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Theorem | sbcbidv 3021* | Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) |
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Theorem | sbcbii 3022 | Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.) |
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Theorem | eqsbc2 3023* | Substitution for the right-hand side in an equality. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.) |
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Theorem | sbc3an 3024 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.) |
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Theorem | sbcel1v 3025* | Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) |
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Theorem | sbcel2gv 3026* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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Theorem | sbcel21v 3027* | Class substitution into a membership relation. One direction of sbcel2gv 3026 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
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Theorem | sbcimdv 3028* | Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1457). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
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Theorem | sbctt 3029 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |
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Theorem | sbcgf 3030 | 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.) |
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Theorem | sbc19.21g 3031 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |
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Theorem | sbcg 3032* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3030. (Contributed by Alan Sare, 10-Nov-2012.) |
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Theorem | sbc2iegf 3033* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |
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Theorem | sbc2ie 3034* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |
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Theorem | sbc2iedv 3035* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
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Theorem | sbc3ie 3036* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) |
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Theorem | sbccomlem 3037* | Lemma for sbccom 3038. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) |
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Theorem | sbccom 3038* | Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
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Theorem | sbcralt 3039* | Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.) |
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Theorem | sbcrext 3040* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
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Theorem | sbcralg 3041* | Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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Theorem | sbcrex 3042* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
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Theorem | sbcreug 3043* | Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.) |
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Theorem | sbcabel 3044* | Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.) |
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Theorem | rspsbc 3045* | 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 1775 and spsbc 2974. See also rspsbca 3046 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
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Theorem | rspsbca 3046* | Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.) |
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Theorem | rspesbca 3047* | Existence form of rspsbca 3046. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
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Theorem | spesbc 3048 | Existence form of spsbc 2974. (Contributed by Mario Carneiro, 18-Nov-2016.) |
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Theorem | spesbcd 3049 | form of spsbc 2974. (Contributed by Mario Carneiro, 9-Feb-2017.) |
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Theorem | sbcth2 3050* | A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
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Theorem | ra5 3051 | 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 1584. (Contributed by NM, 16-Jan-2004.) |
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Theorem | rmo2ilem 3052* | Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.) |
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Theorem | rmo2i 3053* | Condition implying restricted at-most-one quantifier. (Contributed by NM, 17-Jun-2017.) |
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Theorem | rmo3 3054* | Restricted at-most-one quantifier using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) |
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Theorem | rmob 3055* | Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.) |
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Theorem | rmoi 3056* | Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) |
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Syntax | csb 3057 | Extend class notation to include the proper substitution of a class for a set into another class. |
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Definition | df-csb 3058* | 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 2962, to prevent ambiguity. Theorem sbcel1g 3076 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3086 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.) |
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Theorem | csb2 3059* |
Alternate expression for the proper substitution into a class, without
referencing substitution into a wff. Note that ![]() ![]() ![]() |
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Theorem | csbeq1 3060 | Analog of dfsbcq 2964 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
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Theorem | cbvcsbw 3061* | Version of cbvcsb 3062 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
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Theorem | cbvcsb 3062 |
Change bound variables in a class substitution. Interestingly, this
does not require any bound variable conditions on ![]() |
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Theorem | cbvcsbv 3063* | 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.) |
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Theorem | csbeq1d 3064 | Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.) |
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Theorem | csbid 3065 | Analog of sbid 1774 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
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Theorem | csbeq1a 3066 | Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
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Theorem | csbco 3067* |
Composition law for chained substitutions into a class.
Use the weaker csbcow 3068 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.) |
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Theorem | csbcow 3068* | Composition law for chained substitutions into a class. Version of csbco 3067 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 25-Aug-2024.) |
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Theorem | csbtt 3069 |
Substitution doesn't affect a constant ![]() ![]() |
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Theorem | csbconstgf 3070 |
Substitution doesn't affect a constant ![]() ![]() |
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Theorem | csbconstg 3071* |
Substitution doesn't affect a constant ![]() ![]() |
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Theorem | sbcel12g 3072 | Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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Theorem | sbceqg 3073 | Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
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Theorem | sbcnel12g 3074 | Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.) |
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Theorem | sbcne12g 3075 | Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) |
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Theorem | sbcel1g 3076* |
Move proper substitution in and out of a membership relation. Note that
the scope of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sbceq1g 3077* | Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.) |
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Theorem | sbcel2g 3078* | Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) |
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Theorem | sbceq2g 3079* | Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.) |
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Theorem | csbcomg 3080* | Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) |
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Theorem | csbeq2 3081 | Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
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Theorem | csbeq2d 3082 | Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
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Theorem | csbeq2dv 3083* | Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
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Theorem | csbeq2i 3084 | Formula-building inference for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
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Theorem | csbvarg 3085 | The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.) |
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Theorem | sbccsbg 3086* | Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) |
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Theorem | sbccsb2g 3087 | Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) |
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Theorem | nfcsb1d 3088 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfcsb1 3089 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfcsb1v 3090* | Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfsbcdw 3091* | Version of nfsbcd 2982 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
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Theorem | nfcsbd 3092 | Deduction version of nfcsb 3094. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfcsbw 3093* | Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3094 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) |
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Theorem | nfcsb 3094 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
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Theorem | csbhypf 3095* | Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2786 for class substitution version. (Contributed by NM, 19-Dec-2008.) |
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Theorem | csbiebt 3096* | Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3100.) (Contributed by NM, 11-Nov-2005.) |
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Theorem | csbiedf 3097* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.) |
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Theorem | csbieb 3098* |
Bidirectional conversion between an implicit class substitution
hypothesis ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | csbiebg 3099* |
Bidirectional conversion between an implicit class substitution
hypothesis ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | csbiegf 3100* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
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