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Theorem fneq12d 5538
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 5536 . 2  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
3 fneq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43fneq2d 5537 . 2  |-  ( ph  ->  ( G  Fn  A  <->  G  Fn  B ) )
52, 4bitrd 245 1  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    Fn wfn 5449
This theorem is referenced by:  seqfn  11335  sscres  14023  reschomf  14031  funcres  14093  psrvscafval  16454  ressprdsds  18401  fneq12  24046  funcoressn  27967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-fun 5456  df-fn 5457
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