Theorem caovcomd 6026

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 6025	. 2 |- ( ( ph /\ ( A e. S /\ B e. S ) ) -> ( A F B ) = ( B F A ) )
6	1, 2, 3, 5	syl12anc 1236	1 |- ( ph -> ( A F B ) = ( B F A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148  (class class class)co 5870

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-iota 5175  df-fv 5221  df-ov 5873

This theorem is referenced by:  caovcanrd  6033  caovord2d  6039  caovdir2d  6046  caov32d  6050  caov12d  6051  caov31d  6052  caov411d  6055  caov42d  6056  caovimo  6063  ecopovsymg  6629  ecopoverg  6631  ltsonq  7392  prarloclemlo  7488  addextpr  7615  ltsosr  7758  ltasrg  7764  mulextsr1lem  7774  seq3f1olemqsumkj  10491