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Theorem caovcomg 7553
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 3194 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
3 oveq1 7368 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4 oveq2 7369 . . . 4 (𝑥 = 𝐴 → (𝑦𝐹𝑥) = (𝑦𝐹𝐴))
53, 4eqeq12d 2749 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑦𝐹𝑥) ↔ (𝐴𝐹𝑦) = (𝑦𝐹𝐴)))
6 oveq2 7369 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
7 oveq1 7368 . . . 4 (𝑦 = 𝐵 → (𝑦𝐹𝐴) = (𝐵𝐹𝐴))
86, 7eqeq12d 2749 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝑦𝐹𝐴) ↔ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
95, 8rspc2v 3592 . 2 ((𝐴𝑆𝐵𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
102, 9mpan9 508 1 ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  (class class class)co 7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by:  caovcomd  7554  caovcom  7555  caofcom  7656  seqcaopr  13954  cmncom  19588
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