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Theorem caovcom 6784
 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
StepHypRef Expression
1 caovcom.1 . 2 𝐴 ∈ V
2 caovcom.2 . . 3 𝐵 ∈ V
31, 2pm3.2i 471 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 caovcom.3 . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
54a1i 11 . . 3 ((𝐴 ∈ V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
65caovcomg 6782 . 2 ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
71, 3, 6mp2an 707 1 (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  (class class class)co 6604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607 This theorem is referenced by:  caovord2  6799  caov32  6814  caov12  6815  caov42  6820  caovdir  6821  caovmo  6824  ecopovsym  7794  ecopover  7796  ecopoverOLD  7797  genpcl  9774
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