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Theorem caovcom 7326
 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 474 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 caovcom.3 . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
54a1i 11 . . 3 ((𝐴 ∈ V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
65caovcomg 7324 . 2 ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
71, 3, 6mp2an 691 1 (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  (class class class)co 7135 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138 This theorem is referenced by:  caovord2  7341  caov32  7356  caov12  7357  caov42  7362  caovdir  7363  caovmo  7366  ecopovsym  8384  ecopover  8386  genpcl  10421
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