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Theorem fneq12d 6664
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 6662 . 2 (𝜑 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
3 fneq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43fneq2d 6663 . 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 206   = wceq 1537   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-fun 6565  df-fn 6566
This theorem is referenced by:  fneq12  6665  seqfn  14051  sscres  17871  reschomf  17880  funcres  17947  psrvscafval  21986  ressprdsds  24397  rrxmfval  25454  ex-fpar  30491  sseqfn  34372  tfsconcatfn  43328  funcoressn  46992
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