Theorem caovcomd 6211

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

Hypotheses

Ref	Expression
caovcomg.1	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
caovcomd.2	|- ( ph -> A e. S )
caovcomd.3	|- ( ph -> B e. S )

Assertion

Ref	Expression
caovcomd	|- ( ph -> ( A F B ) = ( B F A ) )

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

Proof of Theorem caovcomd

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ph -> ph )
2		caovcomd.2	. 2 |- ( ph -> A e. S )
3		caovcomd.3	. 2 |- ( ph -> B e. S )
4		caovcomg.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
5	4	caovcomg 6210	. 2 |- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )
6	1, 2, 3, 5	syl12anc 1272	1 |- ( ph -> ( A F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203  (class class class)co 6050

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 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  caovcanrd  6218  caovord2d  6224  caovdir2d  6231  caov32d  6235  caov12d  6236  caov31d  6237  caov411d  6240  caov42d  6241  caovimo  6248  ecopovsymg  6868  ecopoverg  6870  ltsonq  7713  prarloclemlo  7809  addextpr  7936  ltsosr  8079  ltasrg  8085  mulextsr1lem  8095  seq3f1olemqsumkj  10873  seqf1oglem2a  10880