Theorem aoveq123d 45876

Description: Equality deduction for operation value, analogous to oveq123d 7429. (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 4881	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
5	1, 4	afveq12d 45831	. 2 ⊢ (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6		df-aov 45819	. 2 ⊢ ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7		df-aov 45819	. 2 ⊢ ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
8	5, 6, 7	3eqtr4g 2797	1 ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Syntax hints:   → wi 4   = wceq 1541  ⟨cop 4634  '''cafv 45815   ((caov 45816

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-aiota 45783  df-dfat 45817  df-afv 45818  df-aov 45819

This theorem is referenced by:  csbaovg  45878  rspceaov  45895  faovcl  45898