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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 257 | . 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 5656 | . 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
7 | 1, 3, 6 | mp2an 402 | 1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 (class class class)co 5512 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: caovord2 5673 caov32 5688 caov12 5689 ecopovsym 6202 ecopover 6204 |
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