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Theorem fneq12d 5337
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 5335 . 2  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
3 fneq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43fneq2d 5336 . 2  |-  ( ph  ->  ( G  Fn  A  <->  G  Fn  B ) )
52, 4bitrd 244 1  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    Fn wfn 5250
This theorem is referenced by:  seqfn  11058  sscres  13700  reschomf  13708  funcres  13770  psrvscafval  16135  ressprdsds  17935  fneq12  23172  funcoressn  27990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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