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Theorem eqvinc 2908
Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
eqvinc.1  |-  A  e. 
_V
Assertion
Ref Expression
eqvinc  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem eqvinc
StepHypRef Expression
1 eqvinc.1 . . . . 5  |-  A  e. 
_V
21isseti 2807 . . . 4  |-  E. x  x  =  A
3 ax-1 5 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  A ) )
4 eqtr 2313 . . . . . . 7  |-  ( ( x  =  A  /\  A  =  B )  ->  x  =  B )
54ex 423 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  B ) )
63, 5jca 518 . . . . 5  |-  ( x  =  A  ->  (
( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
76eximi 1566 . . . 4  |-  ( E. x  x  =  A  ->  E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
8 pm3.43 832 . . . . 5  |-  ( ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  -> 
( A  =  B  ->  ( x  =  A  /\  x  =  B ) ) )
98eximi 1566 . . . 4  |-  ( E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  ->  E. x ( A  =  B  ->  (
x  =  A  /\  x  =  B )
) )
102, 7, 9mp2b 9 . . 3  |-  E. x
( A  =  B  ->  ( x  =  A  /\  x  =  B ) )
111019.37aiv 1853 . 2  |-  ( A  =  B  ->  E. x
( x  =  A  /\  x  =  B ) )
12 eqtr2 2314 . . 3  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1312exlimiv 1624 . 2  |-  ( E. x ( x  =  A  /\  x  =  B )  ->  A  =  B )
1411, 13impbii 180 1  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  eqvincf  2909  tfindsg  4667  findsg  4699  dff13  5799  f1eqcocnv  5821  findcard2s  7115  indpi  8547  dfrdg4  24560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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