Theorem aoveq123d 47804

Description: Equality deduction for operation value, analogous to oveq123d 7432. (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 4850	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
5	1, 4	afveq12d 47759	. 2 ⊢ (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6		df-aov 47747	. 2 ⊢ ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7		df-aov 47747	. 2 ⊢ ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
8	5, 6, 7	3eqtr4g 2829	1 ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Syntax hints:   → wi 4   = wceq 1567  ⟨cop 4600  '''cafv 47743   ((caov 47744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-iota 6493  df-fun 6539  df-fv 6545  df-aiota 47711  df-dfat 47745  df-afv 47746  df-aov 47747

This theorem is referenced by:  csbaovg  47806  rspceaov  47823  faovcl  47826