Theorem caovcomd 6102

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 6101	. 2 |- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )
6	1, 2, 3, 5	syl12anc 1247	1 |- ( ph -> ( A F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175  (class class class)co 5943

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5231  df-fv 5278  df-ov 5946

This theorem is referenced by:  caovcanrd  6109  caovord2d  6115  caovdir2d  6122  caov32d  6126  caov12d  6127  caov31d  6128  caov411d  6131  caov42d  6132  caovimo  6139  ecopovsymg  6720  ecopoverg  6722  ltsonq  7510  prarloclemlo  7606  addextpr  7733  ltsosr  7876  ltasrg  7882  mulextsr1lem  7892  seq3f1olemqsumkj  10654  seqf1oglem2a  10661