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Theorem eqvincg 26006
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 3083 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 ax-1 6 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  A ) )
3 eqtr 2478 . . . . . . 7  |-  ( ( x  =  A  /\  A  =  B )  ->  x  =  B )
43ex 434 . . . . . 6  |-  ( x  =  A  ->  ( A  =  B  ->  x  =  B ) )
52, 4jca 532 . . . . 5  |-  ( x  =  A  ->  (
( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
65eximi 1626 . . . 4  |-  ( E. x  x  =  A  ->  E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) ) )
7 pm3.43 857 . . . . 5  |-  ( ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  -> 
( A  =  B  ->  ( x  =  A  /\  x  =  B ) ) )
87eximi 1626 . . . 4  |-  ( E. x ( ( A  =  B  ->  x  =  A )  /\  ( A  =  B  ->  x  =  B ) )  ->  E. x ( A  =  B  ->  (
x  =  A  /\  x  =  B )
) )
91, 6, 83syl 20 . . 3  |-  ( A  e.  V  ->  E. x
( A  =  B  ->  ( x  =  A  /\  x  =  B ) ) )
10 19.37v 1929 . . 3  |-  ( E. x ( A  =  B  ->  ( x  =  A  /\  x  =  B ) )  <->  ( A  =  B  ->  E. x
( x  =  A  /\  x  =  B ) ) )
119, 10sylib 196 . 2  |-  ( A  e.  V  ->  ( A  =  B  ->  E. x ( x  =  A  /\  x  =  B ) ) )
12 eqtr2 2479 . . 3  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1312exlimiv 1689 . 2  |-  ( E. x ( x  =  A  /\  x  =  B )  ->  A  =  B )
1411, 13impbid1 203 1  |-  ( A  e.  V  ->  ( A  =  B  <->  E. x
( x  =  A  /\  x  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3074
This theorem is referenced by:  funcnv5mpt  26134
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