Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aoveq123d Structured version   Visualization version   GIF version

Theorem aoveq123d 43732
 Description: Equality deduction for operation value, analogous to oveq123d 7156. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
aoveq123d.1 (𝜑𝐹 = 𝐺)
aoveq123d.2 (𝜑𝐴 = 𝐵)
aoveq123d.3 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
aoveq123d (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Proof of Theorem aoveq123d
StepHypRef Expression
1 aoveq123d.1 . . 3 (𝜑𝐹 = 𝐺)
2 aoveq123d.2 . . . 4 (𝜑𝐴 = 𝐵)
3 aoveq123d.3 . . . 4 (𝜑𝐶 = 𝐷)
42, 3opeq12d 4773 . . 3 (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
51, 4afveq12d 43687 . 2 (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩))
6 df-aov 43675 . 2 ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩)
7 df-aov 43675 . 2 ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩)
85, 6, 73eqtr4g 2858 1 (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ⟨cop 4531  '''cafv 43671   ((caov 43672 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-aiota 43640  df-dfat 43673  df-afv 43674  df-aov 43675 This theorem is referenced by:  csbaovg  43734  rspceaov  43751  faovcl  43754
 Copyright terms: Public domain W3C validator