Theorem caovcom 7589

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 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 7587	. 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
7	1, 3, 6	mp2an 692	1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  (class class class)co 7390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393

This theorem is referenced by:  caovord2  7604  caov32  7619  caov12  7620  caov42  7625  caovdir  7626  caovmo  7629  ecopovsym  8795  ecopover  8797  genpcl  10968