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Theorem afveq12d 46410
Description: Equality deduction for function value, analogous to fveq12d 6892. (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 46403 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
41, 2fveq12d 6892 . . 3 (𝜑 → (𝐹𝐴) = (𝐺𝐵))
53, 4ifbieq1d 4547 . 2 (𝜑 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = if(𝐺 defAt 𝐵, (𝐺𝐵), V))
6 dfafv2 46409 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
7 dfafv2 46409 . 2 (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺𝐵), V)
85, 6, 73eqtr4g 2791 1 (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  Vcvv 3468  ifcif 4523  cfv 6537   defAt wdfat 46393  '''cafv 46394
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6489  df-fun 6539  df-fv 6545  df-aiota 46362  df-dfat 46396  df-afv 46397
This theorem is referenced by:  afveq1  46411  afveq2  46412  csbafv12g  46414  afvco2  46453  aoveq123d  46455
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