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Mirrors > Home > MPE Home > Th. List > caovcom | Structured version Visualization version 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 471 | . 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
4 | caovcom.3 | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
6 | 5 | caovcomg 7545 | . 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
7 | 1, 3, 6 | mp2an 690 | 1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 (class class class)co 7353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7356 |
This theorem is referenced by: caovord2 7562 caov32 7577 caov12 7578 caov42 7583 caovdir 7584 caovmo 7587 ecopovsym 8754 ecopover 8756 genpcl 10940 |
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