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Theorem cleqf 2246
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2182. (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 2077 . 2  |-  ( A  =  B  <->  A. y
( y  e.  A  <->  y  e.  B ) )
2 nfv 1462 . . 3  |-  F/ y ( x  e.  A  <->  x  e.  B )
3 cleqf.1 . . . . 5  |-  F/_ x A
43nfcri 2217 . . . 4  |-  F/ x  y  e.  A
5 cleqf.2 . . . . 5  |-  F/_ x B
65nfcri 2217 . . . 4  |-  F/ x  y  e.  B
74, 6nfbi 1522 . . 3  |-  F/ x
( y  e.  A  <->  y  e.  B )
8 eleq1 2145 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
9 eleq1 2145 . . . 4  |-  ( x  =  y  ->  (
x  e.  B  <->  y  e.  B ) )
108, 9bibi12d 233 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  <->  x  e.  B )  <->  ( y  e.  A  <->  y  e.  B
) ) )
112, 7, 10cbval 1679 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  B
)  <->  A. y ( y  e.  A  <->  y  e.  B ) )
121, 11bitr4i 185 1  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   F/_wnfc 2210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212
This theorem is referenced by:  abid2f  2247  n0rf  3277  eq0  3283  iunab  3745  iinab  3760  sniota  4945
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