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Theorem eqriv2 25050
Description: Infer equality of classes from equivalence of membership. (Contributed by FL, 22-Nov-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
eqriv2.1  |-  F/_ x A
eqriv2.2  |-  F/_ x B
eqriv2.3  |-  ( x  e.  A  <->  x  e.  B )
Assertion
Ref Expression
eqriv2  |-  A  =  B

Proof of Theorem eqriv2
StepHypRef Expression
1 eqriv2.1 . . 3  |-  F/_ x A
2 eqriv2.2 . . 3  |-  F/_ x B
31, 2cleqf 2456 . 2  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
4 eqriv2.3 . 2  |-  ( x  e.  A  <->  x  e.  B )
53, 4mpgbir 1540 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   F/_wnfc 2419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-cleq 2289  df-clel 2292  df-nfc 2421
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