Theorem caovord3d

Description: Ordering law. (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 ) )
	caovord3d.5	|- ( ph -> D e. S )
Assertion	caovord3d	|- ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )

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		caovord3d.5	|- ( ph -> D e. S )
7		breq1	|- ( ( A F B ) = ( C F D ) -> ( ( A F B ) R ( C F B ) <-> ( C F D ) R ( C F B ) ) )
8	1 2 4 3 5	caovord2d	|- ( ph -> ( A R C <-> ( A F B ) R ( C F B ) ) )
9	1 6 3 4	caovordd	|- ( ph -> ( D R B <-> ( C F D ) R ( C F B ) ) )
10	8 9	bibi12d	|- ( ph -> ( ( A R C <-> D R B ) <-> ( ( A F B ) R ( C F B ) <-> ( C F D ) R ( C F B ) ) ) )
11	7 10	syl5ibr	|- ( ph -> ( ( A F B ) = ( C F D ) -> ( A R C <-> D R B ) ) )