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Theorem fneq12d 5353
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 5351 . 2  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
3 fneq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43fneq2d 5352 . 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 1632    Fn wfn 5266
This theorem is referenced by:  seqfn  11074  sscres  13716  reschomf  13724  funcres  13786  psrvscafval  16151  ressprdsds  17951  fneq12  23188  funcoressn  28095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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