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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 6046 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
6 | 1, 2, 3, 5 | syl12anc 1246 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2159 (class class class)co 5890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2170 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-rex 2473 df-v 2753 df-un 3147 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-br 4018 df-iota 5192 df-fv 5238 df-ov 5893 |
This theorem is referenced by: caovcanrd 6054 caovord2d 6060 caovdir2d 6067 caov32d 6071 caov12d 6072 caov31d 6073 caov411d 6076 caov42d 6077 caovimo 6084 ecopovsymg 6651 ecopoverg 6653 ltsonq 7414 prarloclemlo 7510 addextpr 7637 ltsosr 7780 ltasrg 7786 mulextsr1lem 7796 seq3f1olemqsumkj 10515 |
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