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Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeueq 2901* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
 |-  ( A  e.  _V  <->  E! x  x  =  A )
 
Theoremeueq1 2902* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |-  A  e.  _V   =>    |-  E! x  x  =  A
 
Theoremeueq2dc 2903* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (DECID 
 ph  ->  E! x ( ( ph  /\  x  =  A )  \/  ( -.  ph  /\  x  =  B ) ) )
 
Theoremeueq3dc 2904* Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  -.  ( ph  /\  ps )   =>    |-  (DECID  ph  ->  (DECID  ps 
 ->  E! x ( (
 ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C )
 ) ) )
 
Theoremmoeq 2905* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)
 |- 
 E* x  x  =  A
 
Theoremmoeq3dc 2906* "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  -.  ( ph  /\  ps )   =>    |-  (DECID  ph  ->  (DECID  ps 
 ->  E* x ( (
 ph  /\  x  =  A )  \/  ( -.  ( ph  \/  ps )  /\  x  =  B )  \/  ( ps  /\  x  =  C )
 ) ) )
 
Theoremmosubt 2907* "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)
 |-  ( A. y E* x ph  ->  E* x E. y ( y  =  A  /\  ph )
 )
 
Theoremmosub 2908* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)
 |- 
 E* x ph   =>    |- 
 E* x E. y
 ( y  =  A  /\  ph )
 
Theoremmo2icl 2909* Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x (
 ph  ->  x  =  A )  ->  E* x ph )
 
Theoremmob2 2910* Consequence of "at most one". (Contributed by NM, 2-Jan-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
 
Theoremmoi2 2911* Consequence of "at most one". (Contributed by NM, 29-Jun-2008.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ( ph  /\  ps )
 )  ->  x  =  A )
 
Theoremmob 2912* Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( ( A  e.  C  /\  B  e.  D )  /\  E* x ph  /\  ps )  ->  ( A  =  B 
 <->  ch ) )
 
Theoremmoi 2913* Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( ( A  e.  C  /\  B  e.  D )  /\  E* x ph  /\  ( ps  /\  ch ) ) 
 ->  A  =  B )
 
Theoremmorex 2914* Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  B  e.  _V   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( E. x  e.  A  ph  /\  E* x ph )  ->  ( ps  ->  B  e.  A ) )
 
Theoremeuxfr2dc 2915* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 14-Nov-2004.)
 |-  A  e.  _V   &    |-  E* y  x  =  A   =>    |-  (DECID  E. y E. x ( x  =  A  /\  ph )  ->  ( E! x E. y ( x  =  A  /\  ph )  <->  E! y ph ) )
 
Theoremeuxfrdc 2916* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 14-Nov-2004.)
 |-  A  e.  _V   &    |-  E! y  x  =  A   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  (DECID 
 E. y E. x ( x  =  A  /\  ps )  ->  ( E! x ph  <->  E! y ps )
 )
 
Theoremeuind 2917* Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)
 |-  B  e.  _V   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  y  ->  A  =  B )   =>    |-  ( ( A. x A. y ( (
 ph  /\  ps )  ->  A  =  B ) 
 /\  E. x ph )  ->  E! z A. x ( ph  ->  z  =  A ) )
 
Theoremreu2 2918* A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
 |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  A. x  e.  A  A. y  e.  A  ( ( ph  /\ 
 [ y  /  x ] ph )  ->  x  =  y ) ) )
 
Theoremreu6 2919* A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)
 |-  ( E! x  e.  A  ph  <->  E. y  e.  A  A. x  e.  A  (
 ph 
 <->  x  =  y ) )
 
Theoremreu3 2920* A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
 |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  E. y  e.  A  A. x  e.  A  ( ph  ->  x  =  y ) ) )
 
Theoremreu6i 2921* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ( B  e.  A  /\  A. x  e.  A  ( ph  <->  x  =  B ) )  ->  E! x  e.  A  ph )
 
Theoremeqreu 2922* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( ( B  e.  A  /\  ps  /\ 
 A. x  e.  A  ( ph  ->  x  =  B ) )  ->  E! x  e.  A  ph )
 
Theoremrmo4 2923* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremreu4 2924* Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  A. x  e.  A  A. y  e.  A  ( ( ph  /\ 
 ps )  ->  x  =  y ) ) )
 
Theoremreu7 2925* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph  /\  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) ) )
 
Theoremreu8 2926* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x  e.  A  ph  <->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  x  =  y ) ) )
 
Theoremrmo3f 2927* 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.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ y ph   =>    |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
Theoremrmo4f 2928* 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.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremreueq 2929* Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
 
Theoremrmoan 2930 Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x  e.  A  ph  ->  E* x  e.  A  ( ps  /\  ph ) )
 
Theoremrmoim 2931 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( E* x  e.  A  ps  ->  E* x  e.  A  ph ) )
 
Theoremrmoimia 2932 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |-  ( E* x  e.  A  ps  ->  E* x  e.  A  ph )
 
Theoremrmoimi2 2933 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |- 
 A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  B  /\  ps )
 )   =>    |-  ( E* x  e.  B  ps  ->  E* x  e.  A  ph )
 
Theorem2reuswapdc 2934* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
 |-  (DECID 
 E. x E. y
 ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  ->  ( A. x  e.  A  E* y  e.  B  ph 
 ->  ( E! x  e.  A  E. y  e.  B  ph  ->  E! y  e.  B  E. x  e.  A  ph ) ) )
 
Theoremreuind 2935* Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  A  =  B )   =>    |-  ( ( A. x A. y ( ( ( A  e.  C  /\  ph )  /\  ( B  e.  C  /\  ps ) )  ->  A  =  B )  /\  E. x ( A  e.  C  /\  ph ) )  ->  E! z  e.  C  A. x ( ( A  e.  C  /\  ph )  ->  z  =  A ) )
 
Theorem2rmorex 2936* Double restricted quantification with "at most one," analogous to 2moex 2105. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( E* x  e.  A  E. y  e.  B  ph  ->  A. y  e.  B  E* x  e.  A  ph )
 
Theoremnelrdva 2937* Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
 |-  ( ( ph  /\  x  e.  A )  ->  x  =/=  B )   =>    |-  ( ph  ->  -.  B  e.  A )
 
2.1.7  Conditional equality (experimental)

This is a very useless definition, which "abbreviates"  ( x  =  y  ->  ph ) as CondEq ( x  =  y  ->  ph ). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation  ( x  =  y  ->  ph ).

This is all used as part of a metatheorem: we want to say that  |-  ( x  =  y  ->  ( ph ( x )  <->  ph ( y ) ) ) and  |-  ( x  =  y  ->  A
( x )  =  A ( y ) ) are provable, for any expressions  ph ( x ) or  A ( x ) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations.

The metatheorem comes with a disjoint variables condition: every variable in  ph ( x ) is assumed disjoint from 
x except  x itself. For such a proof by induction, we must consider each of the possible forms of  ph ( x ). If it is a variable other than  x, then we have CondEq ( x  =  y  ->  A  =  A ) or CondEq ( x  =  y  ->  ( ph  <->  ph ) ), which is provable by cdeqth 2942 and reflexivity. Since we are only working with class and wff expressions, it can't be  x itself in set.mm, but if it was we'd have to also prove CondEq
( x  =  y  ->  x  =  y ) (where set equality is being used on the right).

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  x or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder).

In each of the primitive proofs, we are allowed to assume that  y is disjoint from  ph ( x ) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 2947 and cdeqab 2945.

 
Syntaxwcdeq 2938 Extend wff notation to include conditional equality. This is a technical device used in the proof that 
F/ is the not-free predicate, and that definitions are conservative as a result.
 wff CondEq ( x  =  y 
 ->  ph )
 
Definitiondf-cdeq 2939 Define conditional equality. All the notation to the left of the  <-> is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq x y ph. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  (CondEq ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ph ) )
 
Theoremcdeqi 2940 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( x  =  y 
 ->  ph )   =>    |- CondEq ( x  =  y  -> 
 ph )
 
Theoremcdeqri 2941 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  -> 
 ph )   =>    |-  ( x  =  y 
 ->  ph )
 
Theoremcdeqth 2942 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- CondEq ( x  =  y  -> 
 ph )
 
Theoremcdeqnot 2943 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( -.  ph  <->  -. 
 ps ) )
 
Theoremcdeqal 2944* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( A. z ph  <->  A. z ps )
 )
 
Theoremcdeqab 2945* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  { z  |  ph }  =  {
 z  |  ps }
 )
 
Theoremcdeqal1 2946* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( A. x ph  <->  A. y ps )
 )
 
Theoremcdeqab1 2947* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  { x  |  ph }  =  {
 y  |  ps }
 )
 
Theoremcdeqim 2948 Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   &    |- CondEq ( x  =  y  ->  ( ch 
 <-> 
 th ) )   =>    |- CondEq ( x  =  y  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) )
 
Theoremcdeqcv 2949 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  x  =  y )
 
Theoremcdeqeq 2950 Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  A  =  B )   &    |- CondEq ( x  =  y  ->  C  =  D )   =>    |- CondEq ( x  =  y  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremcdeqel 2951 Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  A  =  B )   &    |- CondEq ( x  =  y  ->  C  =  D )   =>    |- CondEq ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )
 
Theoremnfcdeq 2952* If we have a conditional equality proof, where  ph is  ph ( x ) and  ps is  ph (
y ), and  ph (
x ) in fact does not have  x free in it according to  F/, then  ph ( x )  <->  ph ( y ) unconditionally. This proves that  F/ x ph is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   &    |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( ph  <->  ps )
 
Theoremnfccdeq 2953* Variation of nfcdeq 2952 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  F/_ x A   &    |- CondEq ( x  =  y  ->  A  =  B )   =>    |-  A  =  B
 
2.1.8  Russell's Paradox
 
Theoremru 2954 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 
A  e.  _V, asserted that any collection of sets  A is a set i.e. belongs to the universe 
_V of all sets. In particular, by substituting  { x  |  x  e/  x } (the "Russell class") for  A, it asserted  { x  |  x  e/  x }  e.  _V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove  { x  |  x  e/  x }  e/  _V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom asserting that  A is a set only when it is smaller than some other set  B. The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 4105. (Contributed by NM, 7-Aug-1994.)

 |- 
 { x  |  x  e/  x }  e/  _V
 
2.1.9  Proper substitution of classes for sets
 
Syntaxwsbc 2955 Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class  A for setvar variable  x in wff  ph".
 wff  [. A  /  x ].
 ph
 
Definitiondf-sbc 2956 Define the proper substitution of a class for a set.

When  A is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 2980 for our definition, which always evaluates to true for proper classes.

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 2957 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of  ph, and it does not seem possible to express it with a single closed formula.

If we did not want to commit to any specific proper class behavior, we could use this definition only to prove Theorem dfsbcq 2957, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2956 in the form of sbc8g 2962. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of  A in every use of this definition) we allow direct reference to df-sbc 2956 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

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.)

 |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } )
 
Theoremdfsbcq 2957 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 2956 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 2958 instead of df-sbc 2956. (dfsbcq2 2958 is needed because unlike Quine we do not overload the df-sb 1756 syntax.) As a consequence of these theorems, we can derive sbc8g 2962, which is a weaker version of df-sbc 2956 that leaves substitution undefined when  A is a proper class.

However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2962, so we will allow direct use of df-sbc 2956. 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.)

 |-  ( A  =  B  ->  ( [. A  /  x ]. ph  <->  [. B  /  x ].
 ph ) )
 
Theoremdfsbcq2 2958 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 1756 and substitution for class variables df-sbc 2956. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2957. (Contributed by NM, 31-Dec-2016.)
 |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ].
 ph ) )
 
Theoremsbsbc 2959 Show that df-sb 1756 and df-sbc 2956 are equivalent when the class term  A in df-sbc 2956 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1756 for proofs involving df-sbc 2956. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
 |-  ( [ y  /  x ] ph  <->  [. y  /  x ].
 ph )
 
Theoremsbceq1d 2960 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( [. A  /  x ].
 ps 
 <-> 
 [. B  /  x ].
 ps ) )
 
Theoremsbceq1dd 2961 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  [. A  /  x ]. ps )   =>    |-  ( ph  ->  [. B  /  x ]. ps )
 
Theoremsbc8g 2962 This is the closest we can get to df-sbc 2956 if we start from dfsbcq 2957 (see its comments) and dfsbcq2 2958. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } ) )
 
Theoremsbcex 2963 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.)
 |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
 
Theoremsbceq1a 2964 Equality theorem for class substitution. Class version of sbequ12 1764. (Contributed by NM, 26-Sep-2003.)
 |-  ( x  =  A  ->  ( ph  <->  [. A  /  x ].
 ph ) )
 
Theoremsbceq2a 2965 Equality theorem for class substitution. Class version of sbequ12r 1765. (Contributed by NM, 4-Jan-2017.)
 |-  ( A  =  x 
 ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremspsbc 2966 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 1768 and rspsbc 3037. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( A. x ph  -> 
 [. A  /  x ].
 ph ) )
 
Theoremspsbcd 2967 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 1768 and rspsbc 3037. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  [. A  /  x ]. ps )
 
Theoremsbcth 2968 A substitution into a theorem remains true (when  A is a set). (Contributed by NM, 5-Nov-2005.)
 |-  ph   =>    |-  ( A  e.  V  -> 
 [. A  /  x ].
 ph )
 
Theoremsbcthdv 2969* Deduction version of sbcth 2968. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
 
Theoremsbcid 2970 An identity theorem for substitution. See sbid 1767. (Contributed by Mario Carneiro, 18-Feb-2017.)
 |-  ( [. x  /  x ]. ph  <->  ph )
 
Theoremnfsbc1d 2971 Deduction version of nfsbc1 2972. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x [. A  /  x ]. ps )
 
Theoremnfsbc1 2972 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbc1v 2973* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbcd 2974 Deduction version of nfsbc 2975. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x [. A  /  y ]. ps )
 
Theoremnfsbc 2975 Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x [. A  /  y ]. ph
 
Theoremsbcco 2976* A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ].
 ph )
 
Theoremsbcco2 2977* A composition law for class substitution. Importantly,  x may occur free in the class expression substituted for  A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( x  =  y 
 ->  A  =  B )   =>    |-  ( [. x  /  y ]. [. B  /  x ].
 ph 
 <-> 
 [. A  /  x ].
 ph )
 
Theoremsbc5 2978* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph )
 )
 
Theoremsbc6g 2979* An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
 ) )
 
Theoremsbc6 2980* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
 )
 
Theoremsbc7 2981* An equivalence for class substitution in the spirit of df-clab 2157. Note that  x and  A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( [. A  /  x ]. ph  <->  E. y ( y  =  A  /\  [. y  /  x ]. ph )
 )
 
Theoremcbvsbcw 2982* Version of cbvsbc 2983 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  y ]. ps )
 
Theoremcbvsbc 2983 Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  y ]. ps )
 
Theoremcbvsbcv 2984* Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  y ]. ps )
 
Theoremsbciegft 2985* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2986.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/ x ps  /\ 
 A. x ( x  =  A  ->  ( ph 
 <->  ps ) ) ) 
 ->  ( [. A  /  x ]. ph  <->  ps ) )
 
Theoremsbciegf 2986* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
 
Theoremsbcieg 2987* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
 
Theoremsbcie2g 2988* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2989 avoids a disjointness condition on  x and  A by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 y  =  A  ->  ( ps  <->  ch ) )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ch ) )
 
Theoremsbcie 2989* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  ps )
 
Theoremsbciedf 2990* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   &    |-  F/ x ph   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <->  ch ) )
 
Theoremsbcied 2991* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <->  ch ) )
 
Theoremsbcied2 2992* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A  =  B )   &    |-  (
 ( ph  /\  x  =  B )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <->  ch ) )
 
Theoremelrabsf 2993 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2884 has implicit substitution). The hypothesis specifies that 
x must not be a free variable in  B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x B   =>    |-  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ( A  e.  B  /\  [. A  /  x ].
 ph ) )
 
Theoremeqsbc1 2994* Substitution for the left-hand side in an equality. Class version of eqsb1 2274. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
 
Theoremsbcng 2995 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ].
 ph ) )
 
Theoremsbcimg 2996 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps )  <->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps ) ) )
 
Theoremsbcan 2997 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 )
 
Theoremsbcang 2998 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 ) )
 
Theoremsbcor 2999 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
 
Theoremsbcorg 3000 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
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