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Theorem aoveq123d 41579
Description: Equality deduction for operation value, analogous to oveq123d 6711. (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 4441 . . 3 (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
51, 4afveq12d 41534 . 2 (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6 df-aov 41519 . 2 ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7 df-aov 41519 . 2 ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
85, 6, 73eqtr4g 2710 1 (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  cop 4216  '''cafv 41515   ((caov 41516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-dfat 41517  df-afv 41518  df-aov 41519
This theorem is referenced by:  csbaovg  41581  rspceaov  41598  faovcl  41601
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