Theorem caovcom 7565

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 7563	. 2 ⊢ ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
7	1, 3, 6	mp2an 693	1 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  (class class class)co 7368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371

This theorem is referenced by:  caovord2  7580  caov32  7595  caov12  7596  caov42  7601  caovdir  7602  caovmo  7605  ecopovsym  8768  ecopover  8770  genpcl  10931