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Theorem aoveq123d 42223
Description: Equality deduction for operation value, analogous to oveq123d 6943. (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 4644 . . 3 (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
51, 4afveq12d 42178 . 2 (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6 df-aov 42166 . 2 ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7 df-aov 42166 . 2 ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
85, 6, 73eqtr4g 2839 1 (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  cop 4404  '''cafv 42162   ((caov 42163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-res 5367  df-iota 6099  df-fun 6137  df-fv 6143  df-aiota 42119  df-dfat 42164  df-afv 42165  df-aov 42166
This theorem is referenced by:  csbaovg  42225  rspceaov  42242  faovcl  42245
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