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Theorem afveq12d 46542
Description: Equality deduction for function value, analogous to fveq12d 6909. (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 46535 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
41, 2fveq12d 6909 . . 3 (𝜑 → (𝐹𝐴) = (𝐺𝐵))
53, 4ifbieq1d 4556 . 2 (𝜑 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = if(𝐺 defAt 𝐵, (𝐺𝐵), V))
6 dfafv2 46541 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
7 dfafv2 46541 . 2 (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺𝐵), V)
85, 6, 73eqtr4g 2793 1 (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  Vcvv 3473  ifcif 4532  cfv 6553   defAt wdfat 46525  '''cafv 46526
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  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-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-aiota 46494  df-dfat 46528  df-afv 46529
This theorem is referenced by:  afveq1  46543  afveq2  46544  csbafv12g  46546  afvco2  46585  aoveq123d  46587
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