Theorem caovcomg 7613

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 3207	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
3		oveq1 7424	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4		oveq2 7425	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦𝐹𝑥) = (𝑦𝐹𝐴))
5	3, 4	eqeq12d 2778	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑦𝐹𝑥) ↔ (𝐴𝐹𝑦) = (𝑦𝐹𝐴)))
6		oveq2 7425	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
7		oveq1 7424	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦𝐹𝐴) = (𝐵𝐹𝐴))
8	6, 7	eqeq12d 2778	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝑦𝐹𝐴) ↔ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
9	5, 8	rspc2v 3590	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) = (𝑦𝐹𝑥) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)))
10	2, 9	mpan9 516	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  (class class class)co 7417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420

This theorem is used by:  caovcomd  7614  caovcom  7615  caofcom  7719  seqcaopr  14107  cmncom  19931