Theorem caovcomd 7446

Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovcomg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovcomd.2	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovcomd.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
caovcomd	⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

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

Proof of Theorem caovcomd

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝜑 → 𝜑)
2		caovcomd.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		caovcomd.3	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
4		caovcomg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
5	4	caovcomg 7445	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
6	1, 2, 3, 5	syl12anc 833	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  caovcanrd  7453  caovord2d  7459  caovdir2d  7466  caov32d  7470  caov12d  7471  caov31d  7472  caov411d  7475  caov42d  7476  seqf1olem2a  13689