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Mirrors > Home > MPE Home > Th. List > caovcomd | Structured version Visualization version 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 22 | . 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 7622 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
6 | 1, 2, 3, 5 | syl12anc 835 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 |
This theorem is referenced by: caovcanrd 7630 caovord2d 7636 caovdir2d 7643 caov32d 7647 caov12d 7648 caov31d 7649 caov411d 7652 caov42d 7653 seqf1olem2a 14045 |
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