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Theorem fneq12d 6637
Description: Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
Hypotheses
Ref Expression
fneq12d.1 (𝜑𝐹 = 𝐺)
fneq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
fneq12d (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))

Proof of Theorem fneq12d
StepHypRef Expression
1 fneq12d.1 . . 3 (𝜑𝐹 = 𝐺)
21fneq1d 6635 . 2 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
3 fneq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43fneq2d 6636 . 2 (𝜑 → (𝐺 Fn 𝐴𝐺 Fn 𝐵))
52, 4bitrd 279 1 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533   Fn wfn 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-fun 6538  df-fn 6539
This theorem is referenced by:  fneq12  6638  seqfn  13981  sscres  17776  reschomf  17785  funcres  17852  psrvscafval  21846  ressprdsds  24227  rrxmfval  25284  ex-fpar  30219  sseqfn  33918  tfsconcatfn  42646  funcoressn  46306
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