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Theorem aoveq123d 39801
Description: Equality deduction for operation value, analogous to oveq123d 6446. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
aoveq123d.1 (𝜑𝐹 = 𝐺)
aoveq123d.2 (𝜑𝐴 = 𝐵)
aoveq123d.3 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
aoveq123d (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Proof of Theorem aoveq123d
StepHypRef Expression
1 aoveq123d.1 . . 3 (𝜑𝐹 = 𝐺)
2 aoveq123d.2 . . . 4 (𝜑𝐴 = 𝐵)
3 aoveq123d.3 . . . 4 (𝜑𝐶 = 𝐷)
42, 3opeq12d 4246 . . 3 (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
51, 4afveq12d 39756 . 2 (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6 df-aov 39740 . 2 ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7 df-aov 39740 . 2 ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
85, 6, 73eqtr4g 2573 1 (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  cop 4034  '''cafv 39736   ((caov 39737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-res 4944  df-iota 5653  df-fun 5691  df-fv 5697  df-dfat 39738  df-afv 39739  df-aov 39740
This theorem is referenced by:  csbaovg  39803  rspceaov  39820  faovcl  39823
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