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Mirrors > Home > ILE Home > Th. List > caovcomd | GIF version |
Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovcomg.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
caovcomd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovcomd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
Ref | Expression |
---|---|
caovcomd | ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | caovcomd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | caovcomd.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
4 | caovcomg.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
5 | 4 | caovcomg 5676 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
6 | 1, 2, 3, 5 | syl12anc 1167 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 (class class class)co 5532 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: caovcanrd 5684 caovord2d 5690 caovdir2d 5697 caov32d 5701 caov12d 5702 caov31d 5703 caov411d 5706 caov42d 5707 caovimo 5714 ecopovsymg 6228 ecopoverg 6230 ltsonq 6588 prarloclemlo 6684 addextpr 6811 ltsosr 6941 ltasrg 6947 mulextsr1lem 6956 |
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