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Type | Label | Description |
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Statement | ||
Theorem | elab4g 2901* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |
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Theorem | elab3gf 2902 | Membership in a class abstraction, with a weaker antecedent than elabgf 2894. (Contributed by NM, 6-Sep-2011.) |
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Theorem | elab3g 2903* | Membership in a class abstraction, with a weaker antecedent than elabg 2898. (Contributed by NM, 29-Aug-2006.) |
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Theorem | elab3 2904* | Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |
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Theorem | elrabi 2905* | Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
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Theorem | elrabf 2906 | 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 2907* | Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2909.) (Contributed by Thierry Arnoux, 31-Aug-2017.) |
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Theorem | elrab 2908* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) |
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Theorem | elrab3 2909* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.) |
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Theorem | elrabd 2910* | Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2908. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
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Theorem | elrab2 2911* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |
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Theorem | ralab 2912* | Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | ralrab 2913* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
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Theorem | rexab 2914* | 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 2915* | 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 2916* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | ralrab2 2917* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | rexab2 2918* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | rexrab2 2919* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
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Theorem | abidnf 2920* | 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 2921* |
A deduction theorem for converting the inference ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eqeu 2922* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |
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Theorem | eueq 2923* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |
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Theorem | eueq1 2924* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
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Theorem | eueq2dc 2925* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
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Theorem | eueq3dc 2926* | 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 2927* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |
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Theorem | moeq3dc 2928* | "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.) |
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Theorem | mosubt 2929* | "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.) |
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Theorem | mosub 2930* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
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Theorem | mo2icl 2931* | Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.) |
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Theorem | mob2 2932* | Consequence of "at most one". (Contributed by NM, 2-Jan-2015.) |
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Theorem | moi2 2933* | Consequence of "at most one". (Contributed by NM, 29-Jun-2008.) |
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Theorem | mob 2934* | Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.) |
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Theorem | moi 2935* | Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.) |
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Theorem | morex 2936* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | euxfr2dc 2937* |
Transfer existential uniqueness from a variable ![]() ![]() ![]() |
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Theorem | euxfrdc 2938* |
Transfer existential uniqueness from a variable ![]() ![]() ![]() |
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Theorem | euind 2939* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
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Theorem | reu2 2940* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
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Theorem | reu6 2941* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
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Theorem | reu3 2942* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
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Theorem | reu6i 2943* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | eqreu 2944* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | rmo4 2945* | 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 2946* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
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Theorem | reu7 2947* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
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Theorem | reu8 2948* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
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Theorem | rmo3f 2949* | 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 2950* | 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 2951* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
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Theorem | rmoan 2952 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
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Theorem | rmoim 2953 | 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 2954 | 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 2955 | 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 2956* | 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 2957* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
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Theorem | 2rmorex 2958* | Double restricted quantification with "at most one," analogous to 2moex 2124. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
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Theorem | nelrdva 2959* | 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 2960 |
Extend wff notation to include conditional equality. This is a technical
device used in the proof that ![]() |
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Definition | df-cdeq 2961 |
Define conditional equality. All the notation to the left of the ![]() ![]() ![]() ![]() |
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Theorem | cdeqi 2962 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqri 2963 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqth 2964 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqnot 2965 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqal 2966* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqab 2967* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqal1 2968* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqab1 2969* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqim 2970 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqcv 2971 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqeq 2972 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | cdeqel 2973 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfcdeq 2974* |
If we have a conditional equality proof, where ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfccdeq 2975* | Variation of nfcdeq 2974 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | ru 2976 |
Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension,
expressed in our notation as
In 1908, Zermelo rectified this fatal flaw by replacing Comprehension
with a weaker Subset (or Separation) Axiom asserting that |
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Syntax | wsbc 2977 |
Extend wff notation to include the proper substitution of a class for a
set. Read this notation as "the proper substitution of class ![]() ![]() ![]() |
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Definition | df-sbc 2978 |
Define the proper substitution of a class for a set.
When
Our definition also does not produce the same results as discussed in the
proof of Theorem 6.6 of [Quine] p. 42
(although Theorem 6.6 itself does
hold, as shown by dfsbcq 2979 below). Unfortunately, Quine's definition
requires a recursive syntactical breakdown of
If we did not want to commit to any specific proper class behavior, we
could use this definition only to prove Theorem dfsbcq 2979, which holds
for both our definition and Quine's, and from which we can derive a weaker
version of df-sbc 2978 in the form of sbc8g 2985. However, the behavior of
Quine's definition at proper classes is similarly arbitrary, and for
practical reasons (to avoid having to prove sethood of The related definition df-csb defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.) |
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Theorem | dfsbcq 2979 |
This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds
under both our definition and Quine's, provides us with a weak definition
of the proper substitution of a class for a set. Since our df-sbc 2978 does
not result in the same behavior as Quine's for proper classes, if we
wished to avoid conflict with Quine's definition we could start with this
theorem and dfsbcq2 2980 instead of df-sbc 2978. (dfsbcq2 2980 is needed because
unlike Quine we do not overload the df-sb 1774 syntax.) As a consequence of
these theorems, we can derive sbc8g 2985, which is a weaker version of
df-sbc 2978 that leaves substitution undefined when ![]() However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2985, so we will allow direct use of df-sbc 2978. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.) |
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Theorem | dfsbcq2 2980 | This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1774 and substitution for class variables df-sbc 2978. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2979. (Contributed by NM, 31-Dec-2016.) |
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Theorem | sbsbc 2981 |
Show that df-sb 1774 and df-sbc 2978 are equivalent when the class term ![]() |
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Theorem | sbceq1d 2982 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
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Theorem | sbceq1dd 2983 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
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Theorem | sbceqbid 2984* | Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
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Theorem | sbc8g 2985 | This is the closest we can get to df-sbc 2978 if we start from dfsbcq 2979 (see its comments) and dfsbcq2 2980. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
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Theorem | sbcex 2986 | By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.) |
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Theorem | sbceq1a 2987 | Equality theorem for class substitution. Class version of sbequ12 1782. (Contributed by NM, 26-Sep-2003.) |
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Theorem | sbceq2a 2988 | Equality theorem for class substitution. Class version of sbequ12r 1783. (Contributed by NM, 4-Jan-2017.) |
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Theorem | spsbc 2989 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1786 and rspsbc 3060. (Contributed by NM, 16-Jan-2004.) |
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Theorem | spsbcd 2990 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1786 and rspsbc 3060. (Contributed by Mario Carneiro, 9-Feb-2017.) |
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Theorem | sbcth 2991 |
A substitution into a theorem remains true (when ![]() |
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Theorem | sbcthdv 2992* | Deduction version of sbcth 2991. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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Theorem | sbcid 2993 | An identity theorem for substitution. See sbid 1785. (Contributed by Mario Carneiro, 18-Feb-2017.) |
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Theorem | nfsbc1d 2994 | Deduction version of nfsbc1 2995. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfsbc1 2995 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfsbc1v 2996* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfsbcd 2997 | Deduction version of nfsbc 2998. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | nfsbc 2998 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
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Theorem | sbcco 2999* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
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Theorem | sbcco2 3000* |
A composition law for class substitution. Importantly, ![]() ![]() |
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