Theorem caov31d

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypotheses	caovd.1	|- ( ph -> A e. S )
	caovd.2	|- ( ph -> B e. S )
	caovd.3	|- ( ph -> C e. S )
	caovd.com	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
	caovd.ass	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
Assertion	caov31d	|- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) )

Proof

Step	Hyp	Ref	Expression
1		caovd.1	|- ( ph -> A e. S )
2		caovd.2	|- ( ph -> B e. S )
3		caovd.3	|- ( ph -> C e. S )
4		caovd.com	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
5		caovd.ass	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
6	4 1 3	caovcomd	|- ( ph -> ( A F C ) = ( C F A ) )
7	6	oveq1d	|- ( ph -> ( ( A F C ) F B ) = ( ( C F A ) F B ) )
8	1 2 3 4 5	caov32d	|- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )
9	3 2 1 4 5	caov32d	|- ( ph -> ( ( C F B ) F A ) = ( ( C F A ) F B ) )
10	7 8 9	3eqtr4d	|- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) )