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Theorem caovcomd 7623
Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovcomg.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovcomd.2 (𝜑𝐴𝑆)
caovcomd.3 (𝜑𝐵𝑆)
Assertion
Ref Expression
caovcomd (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦

Proof of Theorem caovcomd
StepHypRef Expression
1 id 22 . 2 (𝜑𝜑)
2 caovcomd.2 . 2 (𝜑𝐴𝑆)
3 caovcomd.3 . 2 (𝜑𝐵𝑆)
4 caovcomg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
54caovcomg 7622 . 2 ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
61, 2, 3, 5syl12anc 835 1 (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  (class class class)co 7426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429
This theorem is referenced by:  caovcanrd  7630  caovord2d  7636  caovdir2d  7643  caov32d  7647  caov12d  7648  caov31d  7649  caov411d  7652  caov42d  7653  seqf1olem2a  14045
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