Theorem caovcom 5880

Description: Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)

Hypotheses

Ref	Expression
caovcom.1	⊢ 𝐴 ∈ V
caovcom.2	⊢ 𝐵 ∈ V
caovcom.3	⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)

Assertion

Ref	Expression
caovcom	⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem caovcom

Step	Hyp	Ref	Expression
1		caovcom.1	. 2 ⊢ 𝐴 ∈ V
2		caovcom.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	pm3.2i 268	. 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 5878	. 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
7	1, 3, 6	mp2an 420	1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)

Syntax hints:   ∧ wa 103   = wceq 1312   ∈ wcel 1461  Vcvv 2655  (class class class)co 5726

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-iota 5044  df-fv 5087  df-ov 5729

This theorem is referenced by:  caovord2  5895  caov32  5910  caov12  5911  ecopovsym  6477  ecopover  6479