Theorem caovcomd 6240

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 6239	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
6	1, 2, 3, 5	syl12anc 1276	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  (class class class)co 6079

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6082

This theorem is referenced by:  caovcanrd  6247  caovord2d  6253  caovdir2d  6260  caov32d  6264  caov12d  6265  caov31d  6266  caov411d  6269  caov42d  6270  caovimo  6277  ecopovsymg  6902  ecopoverg  6904  ltsonq  7759  prarloclemlo  7855  addextpr  7982  ltsosr  8125  ltasrg  8131  mulextsr1lem  8141  seq3f1olemqsumkj  10931  seqf1oglem2a  10938