Theorem caovcomd 6061

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 6060	. 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 1364   e. wcel 2160  (class class class)co 5904

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2758  df-un 3152  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-iota 5203  df-fv 5250  df-ov 5907

This theorem is referenced by:  caovcanrd  6068  caovord2d  6074  caovdir2d  6081  caov32d  6085  caov12d  6086  caov31d  6087  caov411d  6090  caov42d  6091  caovimo  6098  ecopovsymg  6668  ecopoverg  6670  ltsonq  7437  prarloclemlo  7533  addextpr  7660  ltsosr  7803  ltasrg  7809  mulextsr1lem  7819  seq3f1olemqsumkj  10543