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Theorem afveq12d 47083
Description: Equality deduction for function value, analogous to fveq12d 6914. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
afveq12d.1 (𝜑𝐹 = 𝐺)
afveq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
afveq12d (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))

Proof of Theorem afveq12d
StepHypRef Expression
1 afveq12d.1 . . . 4 (𝜑𝐹 = 𝐺)
2 afveq12d.2 . . . 4 (𝜑𝐴 = 𝐵)
31, 2dfateq12d 47076 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
41, 2fveq12d 6914 . . 3 (𝜑 → (𝐹𝐴) = (𝐺𝐵))
53, 4ifbieq1d 4555 . 2 (𝜑 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = if(𝐺 defAt 𝐵, (𝐺𝐵), V))
6 dfafv2 47082 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
7 dfafv2 47082 . 2 (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺𝐵), V)
85, 6, 73eqtr4g 2800 1 (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  Vcvv 3478  ifcif 4531  cfv 6563   defAt wdfat 47066  '''cafv 47067
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070
This theorem is referenced by:  afveq1  47084  afveq2  47085  csbafv12g  47087  afvco2  47126  aoveq123d  47128
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