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Theorem eqvinc 2835
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 2720 . . . 4  |-  E. x  x  =  A
3 ax-1 6 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  A ) )
4 eqtr 2175 . . . . . . 7  |-  ( ( x  =  A  /\  A  =  B )  ->  x  =  B )
54ex 114 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  B ) )
63, 5jca 304 . . . . 5  |-  ( x  =  A  ->  (
( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
76eximi 1580 . . . 4  |-  ( E. x  x  =  A  ->  E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
8 pm3.43 592 . . . . 5  |-  ( ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  -> 
( A  =  B  ->  ( x  =  A  /\  x  =  B ) ) )
98eximi 1580 . . . 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 1655 . 2  |-  ( A  =  B  ->  E. x
( x  =  A  /\  x  =  B ) )
12 eqtr2 2176 . . 3  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1312exlimiv 1578 . 2  |-  ( E. x ( x  =  A  /\  x  =  B )  ->  A  =  B )
1411, 13impbii 125 1  |-  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   _Vcvv 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714
This theorem is referenced by:  eqvincf  2837
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