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Theorem cleqf 2279
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2214. (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 2109 . 2  |-  ( A  =  B  <->  A. y
( y  e.  A  <->  y  e.  B ) )
2 nfv 1491 . . 3  |-  F/ y ( x  e.  A  <->  x  e.  B )
3 cleqf.1 . . . . 5  |-  F/_ x A
43nfcri 2249 . . . 4  |-  F/ x  y  e.  A
5 cleqf.2 . . . . 5  |-  F/_ x B
65nfcri 2249 . . . 4  |-  F/ x  y  e.  B
74, 6nfbi 1551 . . 3  |-  F/ x
( y  e.  A  <->  y  e.  B )
8 eleq1 2177 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
9 eleq1 2177 . . . 4  |-  ( x  =  y  ->  (
x  e.  B  <->  y  e.  B ) )
108, 9bibi12d 234 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  <->  x  e.  B )  <->  ( y  e.  A  <->  y  e.  B
) ) )
112, 7, 10cbval 1710 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  B
)  <->  A. y ( y  e.  A  <->  y  e.  B ) )
121, 11bitr4i 186 1  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463   F/_wnfc 2242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2244
This theorem is referenced by:  abid2f  2280  n0rf  3341  eq0  3347  iunab  3825  iinab  3840  sniota  5073
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