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Theorem fneq12d 6654
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 6652 . 2 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
3 fneq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43fneq2d 6653 . 2 (𝜑 → (𝐺 Fn 𝐴𝐺 Fn 𝐵))
52, 4bitrd 278 1 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533   Fn wfn 6548
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-fun 6555  df-fn 6556
This theorem is referenced by:  fneq12  6655  seqfn  14018  sscres  17813  reschomf  17822  funcres  17889  psrvscafval  21898  ressprdsds  24297  rrxmfval  25354  ex-fpar  30292  sseqfn  34043  tfsconcatfn  42798  funcoressn  46453
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