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Type | Label | Description |
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Statement | ||
Theorem | ra5 3001 | 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.) |
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Theorem | rmo2ilem 3002* | Condition implying restricted at-most-one quantifier. (Contributed by Jim Kingdon, 14-Jul-2018.) |
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Theorem | rmo2i 3003* | Condition implying restricted at-most-one quantifier. (Contributed by NM, 17-Jun-2017.) |
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Theorem | rmo3 3004* | 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 3005* | 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 3006* | 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 3007 | Extend class notation to include the proper substitution of a class for a set into another class. |
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Definition | df-csb 3008* | 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 2913, to prevent ambiguity. Theorem sbcel1g 3026 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3036 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.) |
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Theorem | csb2 3009* |
Alternate expression for the proper substitution into a class, without
referencing substitution into a wff. Note that ![]() ![]() ![]() |
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Theorem | csbeq1 3010 | Analog of dfsbcq 2915 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
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Theorem | cbvcsbw 3011* | Version of cbvcsb 3012 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
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Theorem | cbvcsb 3012 |
Change bound variables in a class substitution. Interestingly, this
does not require any bound variable conditions on ![]() |
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Theorem | cbvcsbv 3013* | 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 3014 | Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.) |
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Theorem | csbid 3015 | Analog of sbid 1748 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
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Theorem | csbeq1a 3016 | Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
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Theorem | csbco 3017* |
Composition law for chained substitutions into a class.
Use the weaker csbcow 3018 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.) |
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Theorem | csbcow 3018* | Composition law for chained substitutions into a class. Version of csbco 3017 with a disjoint variable condition. Although currently the proof is a direct reference to csbco 3017, we expect that the additional distinct variable condition will eventually enable us to remove usage of ax-bndl 1487 here. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
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Theorem | csbtt 3019 |
Substitution doesn't affect a constant ![]() ![]() |
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Theorem | csbconstgf 3020 |
Substitution doesn't affect a constant ![]() ![]() |
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Theorem | csbconstg 3021* |
Substitution doesn't affect a constant ![]() ![]() |
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Theorem | sbcel12g 3022 | 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 3023 | 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 3024 | Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.) |
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Theorem | sbcne12g 3025 | Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) |
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Theorem | sbcel1g 3026* |
Move proper substitution in and out of a membership relation. Note that
the scope of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sbceq1g 3027* | Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.) |
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Theorem | sbcel2g 3028* | Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) |
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Theorem | sbceq2g 3029* | Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.) |
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Theorem | csbcomg 3030* | Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) |
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Theorem | csbeq2 3031 | Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
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Theorem | csbeq2d 3032 | 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 3033* | 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 3034 | 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 3035 | 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 3036* | Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) |
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Theorem | sbccsb2g 3037 | Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) |
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Theorem | nfcsb1d 3038 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfcsb1 3039 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfcsb1v 3040* | 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 | nfcsbd 3041 | Deduction version of nfcsb 3042. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfcsb 3042 | Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.) |
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Theorem | csbhypf 3043* | Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2738 for class substitution version. (Contributed by NM, 19-Dec-2008.) |
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Theorem | csbiebt 3044* | Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3048.) (Contributed by NM, 11-Nov-2005.) |
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Theorem | csbiedf 3045* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.) |
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Theorem | csbieb 3046* |
Bidirectional conversion between an implicit class substitution
hypothesis ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | csbiebg 3047* |
Bidirectional conversion between an implicit class substitution
hypothesis ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | csbiegf 3048* | 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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Theorem | csbief 3049* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
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Theorem | csbie 3050* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) |
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Theorem | csbied 3051* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) |
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Theorem | csbied2 3052* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.) |
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Theorem | csbie2t 3053* | Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3054). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.) |
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Theorem | csbie2 3054* | Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.) |
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Theorem | csbie2g 3055* |
Conversion of implicit substitution to explicit class substitution.
This version of sbcie 2947 avoids a disjointness condition on ![]() ![]() |
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Theorem | sbcnestgf 3056 | Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.) |
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Theorem | csbnestgf 3057 | Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) |
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Theorem | sbcnestg 3058* | Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) |
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Theorem | csbnestg 3059* | Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) |
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Theorem | csbnest1g 3060 | Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) |
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Theorem | csbidmg 3061* | Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) |
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Theorem | sbcco3g 3062* | Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) |
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Theorem | csbco3g 3063* | Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) |
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Theorem | rspcsbela 3064* | Special case related to rspsbc 2995. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
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Theorem | sbnfc2 3065* |
Two ways of expressing "![]() ![]() |
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Theorem | csbabg 3066* | Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | cbvralcsf 3067 |
A more general version of cbvralf 2651 that doesn't require ![]() ![]() ![]() ![]() |
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Theorem | cbvrexcsf 3068 | A more general version of cbvrexf 2652 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.) |
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Theorem | cbvreucsf 3069 | A more general version of cbvreuv 2659 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) |
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Theorem | cbvrabcsf 3070 | A more general version of cbvrab 2687 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.) |
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Theorem | cbvralv2 3071* | 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.) |
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Theorem | cbvrexv2 3072* | 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.) |
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Syntax | cdif 3073 |
Extend class notation to include class difference (read: "![]() ![]() |
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Syntax | cun 3074 |
Extend class notation to include union of two classes (read: "![]() ![]() |
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Syntax | cin 3075 |
Extend class notation to include the intersection of two classes (read:
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Syntax | wss 3076 |
Extend wff notation to include the subclass relation. This is
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Theorem | difjust 3077* | Soundness justification theorem for df-dif 3078. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Definition | df-dif 3078* |
Define class difference, also called relative complement. Definition
5.12 of [TakeutiZaring] p. 20.
Contrast this operation with union
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Theorem | unjust 3079* | Soundness justification theorem for df-un 3080. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Definition | df-un 3080* |
Define the union of two classes. Definition 5.6 of [TakeutiZaring]
p. 16. Contrast this operation with difference ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | injust 3081* | Soundness justification theorem for df-in 3082. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Definition | df-in 3082* |
Define the intersection of two classes. Definition 5.6 of
[TakeutiZaring] p. 16. Contrast
this operation with union
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Theorem | dfin5 3083* | Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.) |
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Theorem | dfdif2 3084* | Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.) |
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Theorem | eldif 3085 | Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.) |
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Theorem | eldifd 3086 | If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3085. (Contributed by David Moews, 1-May-2017.) |
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Theorem | eldifad 3087 | If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3085. (Contributed by David Moews, 1-May-2017.) |
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Theorem | eldifbd 3088 | If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3085. (Contributed by David Moews, 1-May-2017.) |
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Definition | df-ss 3089 |
Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18.
Note that ![]() ![]() ![]() |
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Theorem | dfss 3090 | Variant of subclass definition df-ss 3089. (Contributed by NM, 3-Sep-2004.) |
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Theorem | dfss2 3091* | Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.) |
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Theorem | dfss3 3092* | Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.) |
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Theorem | dfss2f 3093 | Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.) |
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Theorem | dfss3f 3094 | Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.) |
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Theorem | nfss 3095 |
If ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ssel 3096 | Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.) |
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Theorem | ssel2 3097 | Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.) |
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Theorem | sseli 3098 | Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sselii 3099 | Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.) |
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Theorem | sseldi 3100 | Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.) |
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