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Theorem cleqf 2593
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2519. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)
Hypotheses
Ref Expression
cleqf.1  |-  F/_ x A
cleqf.2  |-  F/_ x B
Assertion
Ref Expression
cleqf  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )

Proof of Theorem cleqf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cleqf.1 . . 3  |-  F/_ x A
21nfcrii 2558 . 2  |-  ( y  e.  A  ->  A. x  y  e.  A )
3 cleqf.2 . . 3  |-  F/_ x B
43nfcrii 2558 . 2  |-  ( y  e.  B  ->  A. x  y  e.  B )
52, 4cleqh 2519 1  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186   A.wal 1405    = wceq 1407    e. wcel 1844   F/_wnfc 2552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-cleq 2396  df-clel 2399  df-nfc 2554
This theorem is referenced by:  abid2f  2595  abid2fOLD  2596  n0f  3749  iunab  4319  iinab  4334  mbfposr  22353  mbfinf  22366  itg1climres  22415  bnj1366  29228  bj-rabtrALT  31076  compab  36211
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