Theorem caovcomd 6210

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 19	. 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 6209	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
6	1, 2, 3, 5	syl12anc 1272	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  (class class class)co 6049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052

This theorem is referenced by:  caovcanrd  6217  caovord2d  6223  caovdir2d  6230  caov32d  6234  caov12d  6235  caov31d  6236  caov411d  6239  caov42d  6240  caovimo  6247  ecopovsymg  6867  ecopoverg  6869  ltsonq  7709  prarloclemlo  7805  addextpr  7932  ltsosr  8075  ltasrg  8081  mulextsr1lem  8091  seq3f1olemqsumkj  10869  seqf1oglem2a  10876