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Type | Label | Description |
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Statement | ||
Theorem | rspceaimv 2801* | Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.) |
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Theorem | rspcedeq1vd 2802* | Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2799 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.) |
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Theorem | rspcedeq2vd 2803* | Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2799 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.) |
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Theorem | rspc2 2804* | 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.) |
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Theorem | rspc2gv 2805* | Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.) |
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Theorem | rspc2v 2806* | 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.) |
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Theorem | rspc2va 2807* | 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.) |
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Theorem | rspc2ev 2808* | 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.) |
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Theorem | rspc3v 2809* | 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.) |
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Theorem | rspc3ev 2810* | 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.) |
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Theorem | rspceeqv 2811* | Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.) |
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Theorem | eqvinc 2812* | A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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Theorem | eqvincg 2813* | A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.) |
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Theorem | eqvincf 2814 | A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.) |
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Theorem | alexeq 2815* |
Two ways to express substitution of ![]() ![]() ![]() |
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Theorem | ceqex 2816* | Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) |
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Theorem | ceqsexg 2817* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.) |
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Theorem | ceqsexgv 2818* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) |
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Theorem | ceqsrexv 2819* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.) |
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Theorem | ceqsrexbv 2820* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) |
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Theorem | ceqsrex2v 2821* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.) |
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Theorem | clel2 2822* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
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Theorem | clel3g 2823* | An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.) |
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Theorem | clel3 2824* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
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Theorem | clel4 2825* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
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Theorem | pm13.183 2826* |
Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only ![]() |
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Theorem | rr19.3v 2827* | Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.) |
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Theorem | rr19.28v 2828* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.) |
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Theorem | elabgt 2829* | Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2834.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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Theorem | elabgf 2830 | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | elabf 2831* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | elab 2832* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) |
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Theorem | elabd 2833* |
Explicit demonstration the class ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | elabg 2834* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) |
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Theorem | elab2g 2835* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
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Theorem | elab2 2836* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
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Theorem | elab4g 2837* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |
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Theorem | elab3gf 2838 | Membership in a class abstraction, with a weaker antecedent than elabgf 2830. (Contributed by NM, 6-Sep-2011.) |
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Theorem | elab3g 2839* | Membership in a class abstraction, with a weaker antecedent than elabg 2834. (Contributed by NM, 29-Aug-2006.) |
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Theorem | elab3 2840* | Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |
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Theorem | elrabi 2841* | Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
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Theorem | elrabf 2842 | Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) |
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Theorem | elrab3t 2843* | Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2845.) (Contributed by Thierry Arnoux, 31-Aug-2017.) |
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Theorem | elrab 2844* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) |
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Theorem | elrab3 2845* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.) |
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Theorem | elrabd 2846* | Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2844. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
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Theorem | elrab2 2847* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |
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Theorem | ralab 2848* | Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | ralrab 2849* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | rexab 2850* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) |
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Theorem | rexrab 2851* | Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) |
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Theorem | ralab2 2852* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | ralrab2 2853* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | rexab2 2854* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | rexrab2 2855* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | abidnf 2856* | Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) |
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Theorem | dedhb 2857* |
A deduction theorem for converting the inference ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eqeu 2858* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |
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Theorem | eueq 2859* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |
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Theorem | eueq1 2860* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
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Theorem | eueq2dc 2861* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
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Theorem | eueq3dc 2862* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |
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Theorem | moeq 2863* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |
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Theorem | moeq3dc 2864* | "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.) |
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Theorem | mosubt 2865* | "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.) |
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Theorem | mosub 2866* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
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Theorem | mo2icl 2867* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |
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Theorem | mob2 2868* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |
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Theorem | moi2 2869* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |
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Theorem | mob 2870* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
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Theorem | moi 2871* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
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Theorem | morex 2872* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | euxfr2dc 2873* |
Transfer existential uniqueness from a variable ![]() ![]() ![]() |
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Theorem | euxfrdc 2874* |
Transfer existential uniqueness from a variable ![]() ![]() ![]() |
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Theorem | euind 2875* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
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Theorem | reu2 2876* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
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Theorem | reu6 2877* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
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Theorem | reu3 2878* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
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Theorem | reu6i 2879* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | eqreu 2880* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | rmo4 2881* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
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Theorem | reu4 2882* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
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Theorem | reu7 2883* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
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Theorem | reu8 2884* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
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Theorem | rmo3f 2885* | Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
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Theorem | rmo4f 2886* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
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Theorem | reueq 2887* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
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Theorem | rmoan 2888 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
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Theorem | rmoim 2889 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
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Theorem | rmoimia 2890 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
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Theorem | rmoimi2 2891 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
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Theorem | 2reuswapdc 2892* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
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Theorem | reuind 2893* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | 2rmorex 2894* | Double restricted quantification with "at most one," analogous to 2moex 2086. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
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Theorem | nelrdva 2895* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |
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This is a very useless definition, which "abbreviates"
This is all used as part of a metatheorem: we want to say that
The metatheorem comes with a disjoint variables condition: every variable in
Otherwise, it is a primitive operation applied to smaller expressions. In
these cases, for each setvar variable parameter to the operation, we must
consider if it is equal to
In each of the primitive proofs, we are allowed to assume that | ||
Syntax | wcdeq 2896 |
Extend wff notation to include conditional equality. This is a technical
device used in the proof that ![]() |
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Definition | df-cdeq 2897 |
Define conditional equality. All the notation to the left of the ![]() ![]() ![]() ![]() |
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Theorem | cdeqi 2898 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqri 2899 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqth 2900 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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