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Theorem caovcom 5686
 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 261 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 caovcom.3 . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
54a1i 9 . . 3 ((𝐴 ∈ V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
65caovcomg 5684 . 2 ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
71, 3, 6mp2an 410 1 (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   = wceq 1259   ∈ wcel 1409  Vcvv 2574  (class class class)co 5540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543 This theorem is referenced by:  caovord2  5701  caov32  5716  caov12  5717  ecopovsym  6233  ecopover  6235
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