Theorem tgbtwnexch2

Description: Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgbtwnintr.1	|- ( ph -> A e. P )
	tgbtwnintr.2	|- ( ph -> B e. P )
	tgbtwnintr.3	|- ( ph -> C e. P )
	tgbtwnintr.4	|- ( ph -> D e. P )
	tgbtwnexch2.1	|- ( ph -> B e. ( A I D ) )
	tgbtwnexch2.2	|- ( ph -> C e. ( B I D ) )
Assertion	tgbtwnexch2	|- ( ph -> C e. ( A I D ) )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	|- P = ( Base ` G )
2		tkgeom.d	|- .- = ( dist ` G )
3		tkgeom.i	|- I = ( Itv ` G )
4		tkgeom.g	|- ( ph -> G e. TarskiG )
5		tgbtwnintr.1	|- ( ph -> A e. P )
6		tgbtwnintr.2	|- ( ph -> B e. P )
7		tgbtwnintr.3	|- ( ph -> C e. P )
8		tgbtwnintr.4	|- ( ph -> D e. P )
9		tgbtwnexch2.1	|- ( ph -> B e. ( A I D ) )
10		tgbtwnexch2.2	|- ( ph -> C e. ( B I D ) )
11		simpr	|- ( ( ph /\ B = C ) -> B = C )
12	9	adantr	|- ( ( ph /\ B = C ) -> B e. ( A I D ) )
13	11 12	eqeltrrd	|- ( ( ph /\ B = C ) -> C e. ( A I D ) )
14	4	adantr	|- ( ( ph /\ B =/= C ) -> G e. TarskiG )
15	5	adantr	|- ( ( ph /\ B =/= C ) -> A e. P )
16	6	adantr	|- ( ( ph /\ B =/= C ) -> B e. P )
17	7	adantr	|- ( ( ph /\ B =/= C ) -> C e. P )
18	8	adantr	|- ( ( ph /\ B =/= C ) -> D e. P )
19		simpr	|- ( ( ph /\ B =/= C ) -> B =/= C )
20	10	adantr	|- ( ( ph /\ B =/= C ) -> C e. ( B I D ) )
21	9	adantr	|- ( ( ph /\ B =/= C ) -> B e. ( A I D ) )
22	1 2 3 14 17 16 15 18 20 21	tgbtwnintr	|- ( ( ph /\ B =/= C ) -> B e. ( C I A ) )
23	1 2 3 14 17 16 15 22	tgbtwncom	|- ( ( ph /\ B =/= C ) -> B e. ( A I C ) )
24	1 2 3 14 15 16 17 18 19 23 20	tgbtwnouttr2	|- ( ( ph /\ B =/= C ) -> C e. ( A I D ) )
25	13 24	pm2.61dane	|- ( ph -> C e. ( A I D ) )