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Theorem eqvinc 3007
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 2906 . . . 4  |-  E. x  x  =  A
3 ax-1 5 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  A ) )
4 eqtr 2405 . . . . . . 7  |-  ( ( x  =  A  /\  A  =  B )  ->  x  =  B )
54ex 424 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  B ) )
63, 5jca 519 . . . . 5  |-  ( x  =  A  ->  (
( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
76eximi 1582 . . . 4  |-  ( E. x  x  =  A  ->  E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
8 pm3.43 833 . . . . 5  |-  ( ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  -> 
( A  =  B  ->  ( x  =  A  /\  x  =  B ) ) )
98eximi 1582 . . . 4  |-  ( E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  ->  E. x ( A  =  B  ->  (
x  =  A  /\  x  =  B )
) )
102, 7, 9mp2b 10 . . 3  |-  E. x
( A  =  B  ->  ( x  =  A  /\  x  =  B ) )
111019.37aiv 1912 . 2  |-  ( A  =  B  ->  E. x
( x  =  A  /\  x  =  B ) )
12 eqtr2 2406 . . 3  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1312exlimiv 1641 . 2  |-  ( E. x ( x  =  A  /\  x  =  B )  ->  A  =  B )
1411, 13impbii 181 1  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2900
This theorem is referenced by:  eqvincf  3008  tfindsg  4781  findsg  4813  dff13  5944  f1eqcocnv  5968  findcard2s  7286  indpi  8718  dfrdg4  25514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-v 2902
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