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Theorem afveq12d 39862
 Description: Equality deduction for function value, analogous to fveq12d 6109. (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 39858 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
41, 2fveq12d 6109 . . 3 (𝜑 → (𝐹𝐴) = (𝐺𝐵))
53, 4ifbieq1d 4059 . 2 (𝜑 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = if(𝐺 defAt 𝐵, (𝐺𝐵), V))
6 dfafv2 39861 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
7 dfafv2 39861 . 2 (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺𝐵), V)
85, 6, 73eqtr4g 2669 1 (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Vcvv 3173  ifcif 4036  ‘cfv 5804   defAt wdfat 39842  '''cafv 39843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-dfat 39845  df-afv 39846 This theorem is referenced by:  afveq1  39863  afveq2  39864  csbafv12g  39866  afvco2  39905  aoveq123d  39907
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