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Theorem caovcom 6978
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 447 . 2 (𝐴 ∈ V ∧ 𝐵 ∈ V)
4 caovcom.3 . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
54a1i 11 . . 3 ((𝐴 ∈ V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
65caovcomg 6976 . 2 ((𝐴 ∈ V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
71, 3, 6mp2an 672 1 (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  (class class class)co 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796
This theorem is referenced by:  caovord2  6993  caov32  7008  caov12  7009  caov42  7014  caovdir  7015  caovmo  7018  ecopovsym  8002  ecopover  8004  genpcl  10032
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