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Theorem caovcomg 7622
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 3198 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
3 oveq1 7433 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4 oveq2 7434 . . . 4 (𝑥 = 𝐴 → (𝑦𝐹𝑥) = (𝑦𝐹𝐴))
53, 4eqeq12d 2744 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑦𝐹𝑥) ↔ (𝐴𝐹𝑦) = (𝑦𝐹𝐴)))
6 oveq2 7434 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
7 oveq1 7433 . . . 4 (𝑦 = 𝐵 → (𝑦𝐹𝐴) = (𝐵𝐹𝐴))
86, 7eqeq12d 2744 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝑦𝐹𝐴) ↔ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
95, 8rspc2v 3622 . 2 ((𝐴𝑆𝐵𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
102, 9mpan9 505 1 ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  (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:  caovcomd  7623  caovcom  7624  caofcom  7726  seqcaopr  14044  cmncom  19760
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