Theorem caov42d

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 ) ) )
	caovd.4	|- ( ph -> D e. S )
	caovd.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
Assertion	caov42d	|- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) ) )

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		caovd.4	|- ( ph -> D e. S )
7		caovd.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
8	1 2 3 4 5 6 7	caov4d	|- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )
9	4 2 6	caovcomd	|- ( ph -> ( B F D ) = ( D F B ) )
10	9	oveq2d	|- ( ph -> ( ( A F C ) F ( B F D ) ) = ( ( A F C ) F ( D F B ) ) )
11	8 10	eqtrd	|- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( D F B ) ) )