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Theorem caovcom 5972
 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 270 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 caovcom.3 . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
54a1i 9 . . 3 ((𝐴 ∈ V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
65caovcomg 5970 . 2 ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
71, 3, 6mp2an 423 1 (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1335   ∈ wcel 2128  Vcvv 2712  (class class class)co 5818 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5132  df-fv 5175  df-ov 5821 This theorem is referenced by:  caovord2  5987  caov32  6002  caov12  6003  ecopovsym  6569  ecopover  6571
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