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Theorem caovcomg 7629
Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)
Hypothesis
Ref Expression
caovcomg.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
Assertion
Ref Expression
caovcomg ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦

Proof of Theorem caovcomg
StepHypRef Expression
1 caovcomg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
21ralrimivva 3201 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
3 oveq1 7439 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4 oveq2 7440 . . . 4 (𝑥 = 𝐴 → (𝑦𝐹𝑥) = (𝑦𝐹𝐴))
53, 4eqeq12d 2752 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑦𝐹𝑥) ↔ (𝐴𝐹𝑦) = (𝑦𝐹𝐴)))
6 oveq2 7440 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
7 oveq1 7439 . . . 4 (𝑦 = 𝐵 → (𝑦𝐹𝐴) = (𝐵𝐹𝐴))
86, 7eqeq12d 2752 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝑦𝐹𝐴) ↔ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
95, 8rspc2v 3632 . 2 ((𝐴𝑆𝐵𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
102, 9mpan9 506 1 ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  (class class class)co 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  caovcomd  7630  caovcom  7631  caofcom  7735  seqcaopr  14081  cmncom  19817
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