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Theorem eqvinc 2895
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 2794 . . . 4  |-  E. x  x  =  A
3 ax-1 5 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  A ) )
4 eqtr 2300 . . . . . . 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 1563 . . . 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 1563 . . . 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 1841 . 2  |-  ( A  =  B  ->  E. x
( x  =  A  /\  x  =  B ) )
12 eqtr2 2301 . . 3  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1312exlimiv 1666 . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem is referenced by:  eqvincf  2896  tfindsg  4651  findsg  4683  dff13  5783  f1eqcocnv  5805  findcard2s  7099  indpi  8531  dfrdg4  24488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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