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Theorem aoveq123d 45382
Description: Equality deduction for operation value, analogous to oveq123d 7377. (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 4838 . . 3 (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
51, 4afveq12d 45337 . 2 (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6 df-aov 45325 . 2 ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7 df-aov 45325 . 2 ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
85, 6, 73eqtr4g 2801 1 (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cop 4592  '''cafv 45321   ((caov 45322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-res 5645  df-iota 6448  df-fun 6498  df-fv 6504  df-aiota 45289  df-dfat 45323  df-afv 45324  df-aov 45325
This theorem is referenced by:  csbaovg  45384  rspceaov  45401  faovcl  45404
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