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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 474 | . 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 7403 | . 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
7 | 1, 3, 6 | mp2an 692 | 1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 (class class class)co 7213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-ov 7216 |
This theorem is referenced by: caovord2 7420 caov32 7435 caov12 7436 caov42 7441 caovdir 7442 caovmo 7445 ecopovsym 8501 ecopover 8503 genpcl 10622 |
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