Theorem caovcomg 5782

Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.)

Hypothesis

Ref	Expression
caovcomg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))

Assertion

Ref	Expression
caovcomg	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦

Proof of Theorem caovcomg

Step	Hyp	Ref	Expression
1		caovcomg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
2	1	ralrimivva 2455	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
3		oveq1 5641	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4		oveq2 5642	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦𝐹𝑥) = (𝑦𝐹𝐴))
5	3, 4	eqeq12d 2102	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑦𝐹𝑥) ↔ (𝐴𝐹𝑦) = (𝑦𝐹𝐴)))
6		oveq2 5642	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
7		oveq1 5641	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦𝐹𝐴) = (𝐵𝐹𝐴))
8	6, 7	eqeq12d 2102	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝑦𝐹𝐴) ↔ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
9	5, 8	rspc2v 2733	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
10	2, 9	mpan9 275	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1289   ∈ wcel 1438  ∀wral 2359  (class class class)co 5634

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637

This theorem is referenced by:  caovcomd  5783  caovcom  5784  caovlem2d  5819  caofcom  5860  iseqcaopr  9877