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Theorem cleqf 2639
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2566. (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 . . . 4  |-  F/_ x A
21nfcri 2606 . . 3  |-  F/ x  y  e.  A
32nfri 1810 . 2  |-  ( y  e.  A  ->  A. x  y  e.  A )
4 cleqf.2 . . . 4  |-  F/_ x B
54nfcri 2606 . . 3  |-  F/ x  y  e.  B
65nfri 1810 . 2  |-  ( y  e.  B  ->  A. x  y  e.  B )
73, 6cleqh 2566 1  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   F/_wnfc 2599
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-cleq 2443  df-clel 2446  df-nfc 2601
This theorem is referenced by:  abid2f  2641  abid2fOLD  2642  n0f  3745  iunab  4316  iinab  4331  sniota  5508  mbfposr  21248  mbfinf  21261  itg1climres  21310  compab  29837  rfcnpre1  29881  rfcnpre2  29893  bnj1366  32125  bj-rabtrALT  32735
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