Theorem caovord2d

Description: Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypotheses	caovordg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )
	caovordd.2	|- ( ph -> A e. S )
	caovordd.3	|- ( ph -> B e. S )
	caovordd.4	|- ( ph -> C e. S )
	caovord2d.com	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
Assertion	caovord2d	|- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovordg.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y <-> ( z F x ) R ( z F y ) ) )
2		caovordd.2	|- ( ph -> A e. S )
3		caovordd.3	|- ( ph -> B e. S )
4		caovordd.4	|- ( ph -> C e. S )
5		caovord2d.com	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
6	1 2 3 4	caovordd	|- ( ph -> ( A R B <-> ( C F A ) R ( C F B ) ) )
7	5 4 2	caovcomd	|- ( ph -> ( C F A ) = ( A F C ) )
8	5 4 3	caovcomd	|- ( ph -> ( C F B ) = ( B F C ) )
9	7 8	breq12d	|- ( ph -> ( ( C F A ) R ( C F B ) <-> ( A F C ) R ( B F C ) ) )
10	6 9	bitrd	|- ( ph -> ( A R B <-> ( A F C ) R ( B F C ) ) )