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Theorem afveq12d 47145
Description: Equality deduction for function value, analogous to fveq12d 6913. (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 47138 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
41, 2fveq12d 6913 . . 3 (𝜑 → (𝐹𝐴) = (𝐺𝐵))
53, 4ifbieq1d 4550 . 2 (𝜑 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = if(𝐺 defAt 𝐵, (𝐺𝐵), V))
6 dfafv2 47144 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
7 dfafv2 47144 . 2 (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺𝐵), V)
85, 6, 73eqtr4g 2802 1 (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  Vcvv 3480  ifcif 4525  cfv 6561   defAt wdfat 47128  '''cafv 47129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-aiota 47097  df-dfat 47131  df-afv 47132
This theorem is referenced by:  afveq1  47146  afveq2  47147  csbafv12g  47149  afvco2  47188  aoveq123d  47190
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