Theorem caovdid

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

	Ref	Expression
Hypotheses	caovdig.1	|- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) )
	caovdid.2	|- ( ph -> A e. K )
	caovdid.3	|- ( ph -> B e. S )
	caovdid.4	|- ( ph -> C e. S )
Assertion	caovdid	|- ( ph -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovdig.1	|- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) -> ( x G ( y F z ) ) = ( ( x G y ) H ( x G z ) ) )
2		caovdid.2	|- ( ph -> A e. K )
3		caovdid.3	|- ( ph -> B e. S )
4		caovdid.4	|- ( ph -> C e. S )
5		id	|- ( ph -> ph )
6	1	caovdig	|- ( ( ph /\ ( A e. K /\ B e. S /\ C e. S ) ) -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )
7	5 2 3 4 6	syl13anc	|- ( ph -> ( A G ( B F C ) ) = ( ( A G B ) H ( A G C ) ) )