Theorem caovassd

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

	Ref	Expression
Hypotheses	caovassg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
	caovassd.2	|- ( ph -> A e. S )
	caovassd.3	|- ( ph -> B e. S )
	caovassd.4	|- ( ph -> C e. S )
Assertion	caovassd	|- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )

Proof

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