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Theorem eqvincg 2812
Description: A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
Assertion
Ref Expression
eqvincg  |-  ( A  e.  V  ->  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem eqvincg
StepHypRef Expression
1 elisset 2703 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 ax-1 6 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  A ) )
3 eqtr 2158 . . . . . . 7  |-  ( ( x  =  A  /\  A  =  B )  ->  x  =  B )
43ex 114 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  B ) )
52, 4jca 304 . . . . 5  |-  ( x  =  A  ->  (
( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
65eximi 1580 . . . 4  |-  ( E. x  x  =  A  ->  E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
7 pm3.43 592 . . . . 5  |-  ( ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  -> 
( A  =  B  ->  ( x  =  A  /\  x  =  B ) ) )
87eximi 1580 . . . 4  |-  ( E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  ->  E. x ( A  =  B  ->  (
x  =  A  /\  x  =  B )
) )
91, 6, 83syl 17 . . 3  |-  ( A  e.  V  ->  E. x
( A  =  B  ->  ( x  =  A  /\  x  =  B ) ) )
10 nfv 1509 . . . 4  |-  F/ x  A  =  B
111019.37-1 1653 . . 3  |-  ( E. x ( A  =  B  ->  ( x  =  A  /\  x  =  B ) )  -> 
( A  =  B  ->  E. x ( x  =  A  /\  x  =  B ) ) )
129, 11syl 14 . 2  |-  ( A  e.  V  ->  ( A  =  B  ->  E. x ( x  =  A  /\  x  =  B ) ) )
13 eqtr2 2159 . . 3  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1413exlimiv 1578 . 2  |-  ( E. x ( x  =  A  /\  x  =  B )  ->  A  =  B )
1512, 14impbid1 141 1  |-  ( A  e.  V  ->  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332   E.wex 1469    e. wcel 1481
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691
This theorem is referenced by:  dff13  5675  f1eqcocnv  5698
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