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Theorem fneq12d 5598
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 5596 . 2  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
3 fneq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43fneq2d 5597 . 2  |-  ( ph  ->  ( G  Fn  A  <->  G  Fn  B ) )
52, 4bitrd 253 1  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1399    Fn wfn 5508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-br 4385  df-opab 4443  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-fun 5515  df-fn 5516
This theorem is referenced by:  fneq12  5599  seqfn  12045  sscres  15252  reschomf  15260  funcres  15325  psrvscafval  18179  ressprdsds  20982  rrxmfval  21941  sseqfn  28552  funcoressn  32417
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