Theorem tgbtwnexch3

Description: Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 18-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 )
	tgbtwnexch3.5	|- ( ph -> B e. ( A I C ) )
	tgbtwnexch3.6	|- ( ph -> C e. ( A I D ) )
Assertion	tgbtwnexch3	|- ( ph -> C e. ( B 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		tgbtwnexch3.5	|- ( ph -> B e. ( A I C ) )
10		tgbtwnexch3.6	|- ( ph -> C e. ( A I D ) )
11	1 2 3 4 5 6 7 9	tgbtwncom	|- ( ph -> B e. ( C I A ) )
12	1 2 3 4 5 7 8 10	tgbtwncom	|- ( ph -> C e. ( D I A ) )
13	1 2 3 4 6 7 8 5 11 12	tgbtwnintr	|- ( ph -> C e. ( B I D ) )