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| Mirrors > Home > MPE Home > Th. List > caov12 | Structured version Visualization version GIF version | ||
| Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) | 
| Ref | Expression | 
|---|---|
| caov.1 | ⊢ 𝐴 ∈ V | 
| caov.2 | ⊢ 𝐵 ∈ V | 
| caov.3 | ⊢ 𝐶 ∈ V | 
| caov.com | ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | 
| caov.ass | ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | 
| Ref | Expression | 
|---|---|
| caov12 | ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | caov.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 2 | caov.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 3 | caov.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
| 4 | 1, 2, 3 | caovcom 7630 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴) | 
| 5 | 4 | oveq1i 7441 | . 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐵𝐹𝐴)𝐹𝐶) | 
| 6 | caov.3 | . . 3 ⊢ 𝐶 ∈ V | |
| 7 | caov.ass | . . 3 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | |
| 8 | 1, 2, 6, 7 | caovass 7633 | . 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) | 
| 9 | 2, 1, 6, 7 | caovass 7633 | . 2 ⊢ ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶)) | 
| 10 | 5, 8, 9 | 3eqtr3i 2773 | 1 ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 (class class class)co 7431 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: caov31 7662 caov4 7664 caovmo 7670 distrnq 11001 ltaddnq 11014 ltexnq 11015 1idpr 11069 prlem934 11073 prlem936 11087 mulcmpblnrlem 11110 ltsosr 11134 0idsr 11137 1idsr 11138 recexsrlem 11143 mulgt0sr 11145 axmulass 11197 | 
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