Theorem aoveq123d 41579

Description: Equality deduction for operation value, analogous to oveq123d 6711. (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 4441	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
5	1, 4	afveq12d 41534	. 2 ⊢ (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6		df-aov 41519	. 2 ⊢ ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7		df-aov 41519	. 2 ⊢ ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
8	5, 6, 7	3eqtr4g 2710	1 ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Syntax hints:   → wi 4   = wceq 1523  ⟨cop 4216  '''cafv 41515   ((caov 41516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-dfat 41517  df-afv 41518  df-aov 41519

This theorem is referenced by:  csbaovg  41581  rspceaov  41598  faovcl  41601