Theorem aoveq123d 47366

Description: Equality deduction for operation value, analogous to oveq123d 7377. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
aoveq123d.1	⊢ (𝜑 → 𝐹 = 𝐺)
aoveq123d.2	⊢ (𝜑 → 𝐴 = 𝐵)
aoveq123d.3	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
aoveq123d	⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Proof of Theorem aoveq123d

Step	Hyp	Ref	Expression
1		aoveq123d.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2		aoveq123d.2	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
3		aoveq123d.3	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
4	2, 3	opeq12d 4835	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
5	1, 4	afveq12d 47321	. 2 ⊢ (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6		df-aov 47309	. 2 ⊢ ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7		df-aov 47309	. 2 ⊢ ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
8	5, 6, 7	3eqtr4g 2794	1 ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Syntax hints:   → wi 4   = wceq 1541  ⟨cop 4584  '''cafv 47305   ((caov 47306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-res 5634  df-iota 6446  df-fun 6492  df-fv 6498  df-aiota 47273  df-dfat 47307  df-afv 47308  df-aov 47309

This theorem is referenced by:  csbaovg  47368  rspceaov  47385  faovcl  47388