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Theorem cleqf 2598
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
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 dfcleq 2432 . 2  |-  ( A  =  B  <->  A. y
( y  e.  A  <->  y  e.  B ) )
2 nfv 1630 . . 3  |-  F/ y ( x  e.  A  <->  x  e.  B )
3 cleqf.1 . . . . 5  |-  F/_ x A
43nfcri 2568 . . . 4  |-  F/ x  y  e.  A
5 cleqf.2 . . . . 5  |-  F/_ x B
65nfcri 2568 . . . 4  |-  F/ x  y  e.  B
74, 6nfbi 1857 . . 3  |-  F/ x
( y  e.  A  <->  y  e.  B )
8 eleq1 2498 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
9 eleq1 2498 . . . 4  |-  ( x  =  y  ->  (
x  e.  B  <->  y  e.  B ) )
108, 9bibi12d 314 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  <->  x  e.  B )  <->  ( y  e.  A  <->  y  e.  B
) ) )
112, 7, 10cbval 1983 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  B
)  <->  A. y ( y  e.  A  <->  y  e.  B ) )
121, 11bitr4i 245 1  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   F/_wnfc 2561
This theorem is referenced by:  abid2f  2599  n0f  3638  iunab  4139  iinab  4154  sniota  5447  mbfposr  19546  mbfinf  19559  itg1climres  19608  compab  27622  rfcnpre1  27668  rfcnpre2  27680  bnj1366  29263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-cleq 2431  df-clel 2434  df-nfc 2563
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