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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aoveq123d | Structured version Visualization version GIF version |
Description: Equality deduction for operation value, analogous to oveq123d 7437. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aoveq123d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
aoveq123d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
aoveq123d.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
aoveq123d | ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) ) |
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 4877 | . . 3 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩) |
5 | 1, 4 | afveq12d 46576 | . 2 ⊢ (𝜑 → (𝐹'''⟨𝐴, 𝐶⟩) = (𝐺'''⟨𝐵, 𝐷⟩)) |
6 | df-aov 46564 | . 2 ⊢ ((𝐴𝐹𝐶)) = (𝐹'''⟨𝐴, 𝐶⟩) | |
7 | df-aov 46564 | . 2 ⊢ ((𝐵𝐺𝐷)) = (𝐺'''⟨𝐵, 𝐷⟩) | |
8 | 5, 6, 7 | 3eqtr4g 2790 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ⟨cop 4630 '''cafv 46560 ((caov 46561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-res 5684 df-iota 6495 df-fun 6545 df-fv 6551 df-aiota 46528 df-dfat 46562 df-afv 46563 df-aov 46564 |
This theorem is referenced by: csbaovg 46623 rspceaov 46640 faovcl 46643 |
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