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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreu7 2801* 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 2802* 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 ) ) )
 
Theoremreueq 2803* Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
 
Theoremrmoan 2804 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 2805 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 2806 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 2807 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 2808* 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 2809* 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 2810* Double restricted quantification with "at most one," analogous to 2moex 2031. (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 2811* 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 assumption: 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 2816 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 DV assumptions of cdeqab1 2821 and cdeqab 2819.

 
Syntaxwcdeq 2812 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 2813 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 2814 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( x  =  y 
 ->  ph )   =>    |- CondEq ( x  =  y  -> 
 ph )
 
Theoremcdeqri 2815 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  -> 
 ph )   =>    |-  ( x  =  y 
 ->  ph )
 
Theoremcdeqth 2816 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- CondEq ( x  =  y  -> 
 ph )
 
Theoremcdeqnot 2817 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( -.  ph  <->  -. 
 ps ) )
 
Theoremcdeqal 2818* 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 2819* 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 2820* 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 2821* 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 2822 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 2823 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  x  =  y )
 
Theoremcdeqeq 2824 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 2825 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 2826* 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 2827* Variation of nfcdeq 2826 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 2828 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 3932. (Contributed by NM, 7-Aug-1994.)

 |- 
 { x  |  x  e/  x }  e/  _V
 
2.1.9  Proper substitution of classes for sets
 
Syntaxwsbc 2829 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 2830 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 2854 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 2831 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 2831, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2830 in the form of sbc8g 2836. 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 2830 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 2831 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 2830 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 2832 instead of df-sbc 2830. (dfsbcq2 2832 is needed because unlike Quine we do not overload the df-sb 1690 syntax.) As a consequence of these theorems, we can derive sbc8g 2836, which is a weaker version of df-sbc 2830 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 2836, so we will allow direct use of df-sbc 2830. 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 2832 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 1690 and substitution for class variables df-sbc 2830. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2831. (Contributed by NM, 31-Dec-2016.)
 |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ].
 ph ) )
 
Theoremsbsbc 2833 Show that df-sb 1690 and df-sbc 2830 are equivalent when the class term  A in df-sbc 2830 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1690 for proofs involving df-sbc 2830. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
 |-  ( [ y  /  x ] ph  <->  [. y  /  x ].
 ph )
 
Theoremsbceq1d 2834 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 2835 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 2836 This is the closest we can get to df-sbc 2830 if we start from dfsbcq 2831 (see its comments) and dfsbcq2 2832. (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 2837 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 2838 Equality theorem for class substitution. Class version of sbequ12 1698. (Contributed by NM, 26-Sep-2003.)
 |-  ( x  =  A  ->  ( ph  <->  [. A  /  x ].
 ph ) )
 
Theoremsbceq2a 2839 Equality theorem for class substitution. Class version of sbequ12r 1699. (Contributed by NM, 4-Jan-2017.)
 |-  ( A  =  x 
 ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremspsbc 2840 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 1702 and rspsbc 2910. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( A. x ph  -> 
 [. A  /  x ].
 ph ) )
 
Theoremspsbcd 2841 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 1702 and rspsbc 2910. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  [. A  /  x ]. ps )
 
Theoremsbcth 2842 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 2843* Deduction version of sbcth 2842. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
 
Theoremsbcid 2844 An identity theorem for substitution. See sbid 1701. (Contributed by Mario Carneiro, 18-Feb-2017.)
 |-  ( [. x  /  x ]. ph  <->  ph )
 
Theoremnfsbc1d 2845 Deduction version of nfsbc1 2846. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x [. A  /  x ]. ps )
 
Theoremnfsbc1 2846 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbc1v 2847* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbcd 2848 Deduction version of nfsbc 2849. (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 2849 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 2850* 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 2851* 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 2852* 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 2853* 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 2854* 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 2855* An equivalence for class substitution in the spirit of df-clab 2072. 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 )
 )
 
Theoremcbvsbc 2856 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 2857* 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 2858* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2859.) (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 2859* 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 2860* 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 2861* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2862 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 2862* 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 2863* 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 2864* 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 2865* 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 2866 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2760 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 ) )
 
Theoremeqsbc3 2867* Substitution applied to an atomic wff. Set theory version of eqsb3 2188. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
 
Theoremsbcng 2868 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ].
 ph ) )
 
Theoremsbcimg 2869 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 2870 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 )
 
Theoremsbcang 2871 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 2872 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
 
Theoremsbcorg 2873 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 ) ) )
 
Theoremsbcbig 2874 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps ) ) )
 
Theoremsbcn1 2875 Move negation in and out of class substitution. One direction of sbcng 2868 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
 |-  ( [. A  /  x ].  -.  ph  ->  -.  [. A  /  x ].
 ph )
 
Theoremsbcim1 2876 Distribution of class substitution over implication. One direction of sbcimg 2869 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
 |-  ( [. A  /  x ]. ( ph  ->  ps )  ->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps ) )
 
Theoremsbcbi1 2877 Distribution of class substitution over biconditional. One direction of sbcbig 2874 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
 |-  ( [. A  /  x ]. ( ph  <->  ps )  ->  ( [. A  /  x ].
 ph 
 <-> 
 [. A  /  x ].
 ps ) )
 
Theoremsbcbi2 2878 Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( [. A  /  x ].
 ph 
 <-> 
 [. A  /  x ].
 ps ) )
 
Theoremsbcal 2879* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph )
 
Theoremsbcalg 2880* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
 
Theoremsbcex2 2881* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
 |-  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph )
 
Theoremsbcexg 2882* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
 
Theoremsbceqal 2883* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
 |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B )
 )
 
Theoremsbeqalb 2884* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
 |-  ( A  e.  V  ->  ( ( A. x ( ph  <->  x  =  A )  /\  A. x (
 ph 
 <->  x  =  B ) )  ->  A  =  B ) )
 
Theoremsbcbid 2885 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. A  /  x ].
 ch ) )
 
Theoremsbcbidv 2886* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. A  /  x ].
 ch ) )
 
Theoremsbcbii 2887 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
 |-  ( ph  <->  ps )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps )
 
Theoremeqsbc3r 2888* eqsbc3 2867 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  x  <->  B  =  A )
 )
 
Theoremsbc3an 2889 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
 |-  ( [. A  /  x ]. ( ph  /\  ps  /\ 
 ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) )
 
Theoremsbcel1v 2890* Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
 |-  ( [. A  /  x ]. x  e.  B  <->  A  e.  B )
 
Theoremsbcel2gv 2891* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
 
Theoremsbcel21v 2892* Class substitution into a membership relation. One direction of sbcel2gv 2891 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
 |-  ( [. B  /  x ]. A  e.  x  ->  A  e.  B )
 
Theoremsbcimdv 2893* Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1389). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps  ->  [. A  /  x ].
 ch ) )
 
Theoremsbctt 2894 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/ x ph )  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbcgf 2895 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.)
 |- 
 F/ x ph   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc19.21g 2896 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
 |- 
 F/ x ph   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps )  <->  ( ph  ->  [. A  /  x ]. ps ) ) )
 
Theoremsbcg 2897* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2895. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc2iegf 2898* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |- 
 F/ x ps   &    |-  F/ y ps   &    |-  F/ x  B  e.  W   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( [. A  /  x ].
 [. B  /  y ]. ph  <->  ps ) )
 
Theoremsbc2ie 2899* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
 
Theoremsbc2iedv 2900* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [. A  /  x ]. [. B  /  y ]. ps  <->  ch ) )
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