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Theorem cleqf 2344
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2277. (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 2171 . 2  |-  ( A  =  B  <->  A. y
( y  e.  A  <->  y  e.  B ) )
2 nfv 1528 . . 3  |-  F/ y ( x  e.  A  <->  x  e.  B )
3 cleqf.1 . . . . 5  |-  F/_ x A
43nfcri 2313 . . . 4  |-  F/ x  y  e.  A
5 cleqf.2 . . . . 5  |-  F/_ x B
65nfcri 2313 . . . 4  |-  F/ x  y  e.  B
74, 6nfbi 1589 . . 3  |-  F/ x
( y  e.  A  <->  y  e.  B )
8 eleq1 2240 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
9 eleq1 2240 . . . 4  |-  ( x  =  y  ->  (
x  e.  B  <->  y  e.  B ) )
108, 9bibi12d 235 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  <->  x  e.  B )  <->  ( y  e.  A  <->  y  e.  B
) ) )
112, 7, 10cbval 1754 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  B
)  <->  A. y ( y  e.  A  <->  y  e.  B ) )
121, 11bitr4i 187 1  |-  ( A  =  B  <->  A. x
( x  e.  A  <->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148   F/_wnfc 2306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308
This theorem is referenced by:  abid2f  2345  n0rf  3435  eq0  3441  iunab  3930  iinab  3945  sniota  5203
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