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Theorem fneq12d 6598
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 6596 . 2 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
3 fneq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43fneq2d 6597 . 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 1542   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6499  df-fn 6500
This theorem is referenced by:  fneq12  6599  seqfn  13924  sscres  17711  reschomf  17720  funcres  17787  psrvscafval  21374  ressprdsds  23740  rrxmfval  24786  ex-fpar  29448  sseqfn  33047  funcoressn  45362
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