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Mirrors > Home > ILE Home > Th. List > caovcom | GIF version |
Description: Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.) |
Ref | Expression |
---|---|
caovcom.1 | ⊢ 𝐴 ∈ V |
caovcom.2 | ⊢ 𝐵 ∈ V |
caovcom.3 | ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) |
Ref | Expression |
---|---|
caovcom | ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovcom.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | caovcom.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | pm3.2i 268 | . 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
4 | caovcom.3 | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
5 | 4 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
6 | 5 | caovcomg 5878 | . 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
7 | 1, 3, 6 | mp2an 420 | 1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1312 ∈ wcel 1461 Vcvv 2655 (class class class)co 5726 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-iota 5044 df-fv 5087 df-ov 5729 |
This theorem is referenced by: caovord2 5895 caov32 5910 caov12 5911 ecopovsym 6477 ecopover 6479 |
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