Theorem endofsegidand

Description: Deduction form of endofsegid . (Contributed by Scott Fenton, 15-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)

	Ref	Expression
Hypotheses	endofsegidand.1	|- ( ph -> N e. NN )
	endofsegidand.2	|- ( ph -> A e. ( EE ` N ) )
	endofsegidand.3	|- ( ph -> B e. ( EE ` N ) )
	endofsegidand.4	|- ( ph -> C e. ( EE ` N ) )
	endofsegidand.5	|- ( ( ph /\ ps ) -> C Btwn <. A , B >. )
	endofsegidand.6	|- ( ( ph /\ ps ) -> <. A , B >. Cgr <. A , C >. )
Assertion	endofsegidand	|- ( ( ph /\ ps ) -> B = C )

Proof

Step	Hyp	Ref	Expression
1		endofsegidand.1	|- ( ph -> N e. NN )
2		endofsegidand.2	|- ( ph -> A e. ( EE ` N ) )
3		endofsegidand.3	|- ( ph -> B e. ( EE ` N ) )
4		endofsegidand.4	|- ( ph -> C e. ( EE ` N ) )
5		endofsegidand.5	|- ( ( ph /\ ps ) -> C Btwn <. A , B >. )
6		endofsegidand.6	|- ( ( ph /\ ps ) -> <. A , B >. Cgr <. A , C >. )
7		endofsegid	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , B >. Cgr <. A , C >. ) -> B = C ) )
8	1 2 4 3 7	syl13anc	|- ( ph -> ( ( C Btwn <. A , B >. /\ <. A , B >. Cgr <. A , C >. ) -> B = C ) )
9	8	adantr	|- ( ( ph /\ ps ) -> ( ( C Btwn <. A , B >. /\ <. A , B >. Cgr <. A , C >. ) -> B = C ) )
10	5 6 9	mp2and	|- ( ( ph /\ ps ) -> B = C )