Theorem caov12d

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	caov12d	|- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )

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 2	caovcomd	|- ( ph -> ( A F B ) = ( B F A ) )
7	6	oveq1d	|- ( ph -> ( ( A F B ) F C ) = ( ( B F A ) F C ) )
8	5 1 2 3	caovassd	|- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9	5 2 1 3	caovassd	|- ( ph -> ( ( B F A ) F C ) = ( B F ( A F C ) ) )
10	7 8 9	3eqtr3d	|- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )