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Theorem eqvinc 2740
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 2627 . . . 4  |-  E. x  x  =  A
3 ax-1 5 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  A ) )
4 eqtr 2105 . . . . . . 7  |-  ( ( x  =  A  /\  A  =  B )  ->  x  =  B )
54ex 113 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  B ) )
63, 5jca 300 . . . . 5  |-  ( x  =  A  ->  (
( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
76eximi 1536 . . . 4  |-  ( E. x  x  =  A  ->  E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
8 pm3.43 569 . . . . 5  |-  ( ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  -> 
( A  =  B  ->  ( x  =  A  /\  x  =  B ) ) )
98eximi 1536 . . . 4  |-  ( E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  ->  E. x ( A  =  B  ->  (
x  =  A  /\  x  =  B )
) )
102, 7, 9mp2b 8 . . 3  |-  E. x
( A  =  B  ->  ( x  =  A  /\  x  =  B ) )
111019.37aiv 1610 . 2  |-  ( A  =  B  ->  E. x
( x  =  A  /\  x  =  B ) )
12 eqtr2 2106 . . 3  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1312exlimiv 1534 . 2  |-  ( E. x ( x  =  A  /\  x  =  B )  ->  A  =  B )
1411, 13impbii 124 1  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  eqvincf  2742
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