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Theorem List for Metamath Proof Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreueq 2901* Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
 
Theorem2reuswap 2902* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 14-Nov-2004.)
 |-  ( A. x  e.  A  E* y ( y  e.  A  /\  ph )  ->  ( E! x  e.  A  E. y  e.  A  ph  ->  E! y  e.  A  E. x  e.  A  ph ) )
 
Theoremreuind 2903* 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 ) )
 
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 3-ary 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 2908 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 set 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 2913 and cdeqab 2911.

 
Syntaxwcdeq 2904 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 2905 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 2906 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( x  =  y 
 ->  ph )   =>    |- CondEq ( x  =  y  -> 
 ph )
 
Theoremcdeqri 2907 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  -> 
 ph )   =>    |-  ( x  =  y 
 ->  ph )
 
Theoremcdeqth 2908 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- CondEq ( x  =  y  -> 
 ph )
 
Theoremcdeqnot 2909 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( -.  ph  <->  -. 
 ps ) )
 
Theoremcdeqal 2910* 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 2911* 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 2912* 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 2913* 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 2914 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 2915 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  x  =  y )
 
Theoremcdeqeq 2916 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 2917 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 2918* 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 2919* Variation of nfcdeq 2918 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 2920 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 ssex 4055 asserting that  A is a set only when it is smaller than some other set  B. However, Zermelo was then faced with a "chicken and egg" problem of how to show  B is a set, leading him to introduce the set-building axioms of Null Set 0ex 4047, Pairing prex 4111, Union uniex 4407, Power Set pwex 4087, and Infinity omex 7228 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 5187 (whose modern formalization is due to Skolem, also in 1922). Thus in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics!

Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than set variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287).

Russell himself continued in a different direction, avoiding the paradox with his "theory of types." Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 7986 and Cantor's Theorem canth 6178 are provably false! (See ncanth 6179 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 4038 replaces ax-rep 4028) with ax-sep 4038 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic," J. Symb. Logic 9:1-19 (1944).

Under our ZF set theory, every set is a member of the Russell class by elirrv 7195 (derived from the Axiom of Regularity), so for us the Russell class equals the universe 
_V (theorem ruv 7198). See ruALT 7199 for an alternate proof of ru 2920 derived from that fact. (Contributed by NM, 7-Aug-1994.)

 |- 
 { x  |  x  e/  x }  e/  _V
 
2.1.9  Proper substitution of classes for sets
 
Syntaxwsbc 2921 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 set variable  x in wff  ph."
 wff  [. A  /  x ].
 ph
 
Definitiondf-sbc 2922 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 2947 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 2923 below). For example, if  A is a proper class, Quine's substitution of 
A for  y in  0  e.  y evaluates to  0  e.  A rather than our falsehood. (This can be seen by substituting  A,  y, and  0 for for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) 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 2923, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2922 in the form of sbc8g 2928. 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 2922 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 2929 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2923.

The related definition df-csb 3010 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 2923 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 2922 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 2924 instead of df-sbc 2922. (dfsbcq2 2924 is needed because unlike Quine we do not overload the df-sb 1883 syntax.) As a consequence of these theorems, we can derive sbc8g 2928, which is a weaker version of df-sbc 2922 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 2928, so we will allow direct use of df-sbc 2922 after theorem sbc2or 2929 below. 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 2924 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 1883 and substitution for class variables df-sbc 2922. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2923. (Contributed by NM, 31-Dec-2016.)
 |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ].
 ph ) )
 
Theoremsbsbc 2925 Show that df-sb 1883 and df-sbc 2922 are equivalent when the class term  A in df-sbc 2922 is a set variable. This theorem lets us reuse theorems based on df-sb 1883 for proofs involving df-sbc 2922. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
 |-  ( [ y  /  x ] ph  <->  [. y  /  x ].
 ph )
 
Theoremsbceq1d 2926 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( [. A  /  x ].
 ph 
 <-> 
 [. B  /  x ].
 ph ) )
 
Theoremsbceq1dd 2927 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  [. A  /  x ]. ph )   =>    |-  ( ph  ->  [. B  /  x ]. ph )
 
Theoremsbc8g 2928 This is the closest we can get to df-sbc 2922 if we start from dfsbcq 2923 (see its comments) and dfsbcq2 2924. (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 } ) )
 
Theoremsbc2or 2929* The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for  [ A  /  x ] ph behavior at proper classes, matching the sbc5 2945 (false) and sbc6 2947 (true) conclusions. This is interesting since dfsbcq 2923 and dfsbcq2 2924 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem doesn't tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable  y that  ph or  A. may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.)
 |-  ( ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph )
 )  \/  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
 ) )
 
Theoremsbcex 2930 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 2931 Equality theorem for class substitution. Class version of sbequ12 1892. (Contributed by NM, 26-Sep-2003.)
 |-  ( x  =  A  ->  ( ph  <->  [. A  /  x ].
 ph ) )
 
Theoremsbceq2a 2932 Equality theorem for class substitution. Class version of sbequ12r 1893. (Contributed by NM, 4-Jan-2017.)
 |-  ( A  =  x 
 ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theorema4sbc 2933 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 1896 and ra4sbc 2999. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( A. x ph  -> 
 [. A  /  x ].
 ph ) )
 
Theorema4sbcd 2934 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 1896 and ra4sbc 2999. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  [. A  /  x ]. ps )
 
Theoremsbcth 2935 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 2936* Deduction version of sbcth 2935. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
 
Theoremsbcid 2937 An identity theorem for substitution. See sbid 1895. (Contributed by Mario Carneiro, 18-Feb-2017.)
 |-  ( [. x  /  x ]. ph  <->  ph )
 
Theoremnfsbc1d 2938 Deduction version of nfsbc1 2939. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x [. A  /  x ]. ps )
 
Theoremnfsbc1 2939 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbc1v 2940* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbcd 2941 Deduction version of nfsbc 2942. (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 2942 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 2943* 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 2944* 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 2945* 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 2946* 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 2947* 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 2948* An equivalence for class substitution in the spirit of df-clab 2240. 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 2949 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 2950* 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 2951* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2952.) (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 2952* 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 2953* 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 2954* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2955 avoids a disjointness condition on  x ,  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 2955* 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 2956* 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 2957* 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 2958* 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 2959 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2859 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 2960* Substitution applied to an atomic wff. Set theory version of eqsb3 2350. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
 
Theoremsbcng 2961 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ].
 ph ) )
 
Theoremsbcimg 2962 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 2963 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 )
 
Theoremsbcang 2964 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 2965 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
 
Theoremsbcorg 2966 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 2967 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 ) ) )
 
Theoremsbcal 2968* 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 2969* 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 2970* 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 2971* 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 2972* Set theory version of sbeqal1 26763. (Contributed by Andrew Salmon, 28-Jun-2011.)
 |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B )
 )
 
Theoremsbeqalb 2973* 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 2974 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 2975* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. A  /  x ].
 ch ) )
 
Theoremsbcbii 2976 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
 |-  ( ph  <->  ps )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps )
 
TheoremsbcbiiOLD 2977 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.)
 |-  ( ph  <->  ps )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps ) )
 
Theoremeqsbc3r 2978* eqsbc3 2960 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 27306 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A )
 )
 
Theoremsbc3ang 2979 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps  /\ 
 ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) ) )
 
Theoremsbcel1gv 2980* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
 
Theoremsbcel2gv 2981* 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 ) )
 
Theoremsbcimdv 2982* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\  A  e.  V ) 
 ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
 )
 
Theoremsbctt 2983 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 2984 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 2985 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 2986* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2984. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc2iegf 2987* 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 2988* 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 2989* 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 ) )
 
Theoremsbc3ie 2990* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  (
 ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps ) )   =>    |-  ( [. A  /  x ]. [. B  /  y ]. [. C  /  z ]. ph  <->  ps )
 
Theoremsbccomlem 2991* Lemma for sbccom 2992. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom 2992* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbcralt 2993* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
 |-  ( ( A  e.  V  /\  F/_ y A ) 
 ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcrext 2994* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/_ y A ) 
 ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcralg 2995* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcrexg 2996* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcreug 2997* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcabel 2998* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
 |-  F/_ x B   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  | 
 ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B ) )
 
Theoremra4sbc 2999* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 1896 and a4sbc 2933. See also ra4sbca 3000 and ra4csbela 3068. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  [. A  /  x ]. ph )
 )
 
Theoremra4sbca 3000* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  [. A  /  x ]. ph )
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