Theorem aoveq123d 48247

Description: Equality deduction for operation value, analogous to oveq123d 7441. (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 4841	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
5	1, 4	afveq12d 48202	. 2 ⊢ (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6		df-aov 48190	. 2 ⊢ ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7		df-aov 48190	. 2 ⊢ ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
8	5, 6, 7	3eqtr4g 2821	1 ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⟨cop 4590  '''cafv 48186   ((caov 48187

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-aiota 48154  df-dfat 48188  df-afv 48189  df-aov 48190

This theorem is used by:  csbaovg  48249  rspceaov  48266  faovcl  48269