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Theorem eqri 23947
Description: Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
Hypotheses
Ref Expression
eqri.1  |-  F/_ x A
eqri.2  |-  F/_ x B
eqri.3  |-  ( x  e.  A  <->  x  e.  B )
Assertion
Ref Expression
eqri  |-  A  =  B

Proof of Theorem eqri
StepHypRef Expression
1 nftru 1560 . . 3  |-  F/ x  T.
2 eqri.1 . . 3  |-  F/_ x A
3 eqri.2 . . 3  |-  F/_ x B
4 eqri.3 . . . 4  |-  ( x  e.  A  <->  x  e.  B )
54a1i 11 . . 3  |-  (  T. 
->  ( x  e.  A  <->  x  e.  B ) )
61, 2, 3, 5eqrd 3326 . 2  |-  (  T. 
->  A  =  B
)
76trud 1329 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 177    T. wtru 1322    = wceq 1649    e. wcel 1721   F/_wnfc 2527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-in 3287  df-ss 3294
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