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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 470 | . 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 6727 | . 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
7 | 1, 3, 6 | mp2an 704 | 1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: caovord2 6744 caov32 6759 caov12 6760 caov42 6765 caovdir 6766 caovmo 6769 ecopovsym 7736 ecopover 7738 ecopoverOLD 7739 genpcl 9709 |
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