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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 6570. (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 4348 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) |
5 | 1, 4 | afveq12d 39862 | . 2 ⊢ (𝜑 → (𝐹'''〈𝐴, 𝐶〉) = (𝐺'''〈𝐵, 𝐷〉)) |
6 | df-aov 39847 | . 2 ⊢ ((𝐴𝐹𝐶)) = (𝐹'''〈𝐴, 𝐶〉) | |
7 | df-aov 39847 | . 2 ⊢ ((𝐵𝐺𝐷)) = (𝐺'''〈𝐵, 𝐷〉) | |
8 | 5, 6, 7 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 〈cop 4131 '''cafv 39843 ((caov 39844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-dfat 39845 df-afv 39846 df-aov 39847 |
This theorem is referenced by: csbaovg 39909 rspceaov 39926 faovcl 39929 |
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