Theorem caovdird

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

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

Proof

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