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Theorem fneq12d 5304
Description: Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
Hypotheses
Ref Expression
fneq12d.1  |-  ( ph  ->  F  =  G )
fneq12d.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
fneq12d  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  B ) )

Proof of Theorem fneq12d
StepHypRef Expression
1 fneq12d.1 . . 3  |-  ( ph  ->  F  =  G )
21fneq1d 5302 . 2  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
3 fneq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43fneq2d 5303 . 2  |-  ( ph  ->  ( G  Fn  A  <->  G  Fn  B ) )
52, 4bitrd 188 1  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    Fn wfn 5207
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-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-fun 5214  df-fn 5215
This theorem is referenced by:  fneq12  5305  tfrlemi1  6327
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