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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 6943. (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 4644 | . . 3 ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) |
5 | 1, 4 | afveq12d 42178 | . 2 ⊢ (𝜑 → (𝐹'''〈𝐴, 𝐶〉) = (𝐺'''〈𝐵, 𝐷〉)) |
6 | df-aov 42166 | . 2 ⊢ ((𝐴𝐹𝐶)) = (𝐹'''〈𝐴, 𝐶〉) | |
7 | df-aov 42166 | . 2 ⊢ ((𝐵𝐺𝐷)) = (𝐺'''〈𝐵, 𝐷〉) | |
8 | 5, 6, 7 | 3eqtr4g 2839 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 〈cop 4404 '''cafv 42162 ((caov 42163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-int 4711 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-res 5367 df-iota 6099 df-fun 6137 df-fv 6143 df-aiota 42119 df-dfat 42164 df-afv 42165 df-aov 42166 |
This theorem is referenced by: csbaovg 42225 rspceaov 42242 faovcl 42245 |
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