Theorem tgbtwncomb

Description: Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgbtwntriv2.1	|- ( ph -> A e. P )
	tgbtwntriv2.2	|- ( ph -> B e. P )
	tgbtwncomb.3	|- ( ph -> C e. P )
Assertion	tgbtwncomb	|- ( ph -> ( B e. ( A I C ) <-> B e. ( C I A ) ) )

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		tgbtwntriv2.1	|- ( ph -> A e. P )
6		tgbtwntriv2.2	|- ( ph -> B e. P )
7		tgbtwncomb.3	|- ( ph -> C e. P )
8	4	adantr	|- ( ( ph /\ B e. ( A I C ) ) -> G e. TarskiG )
9	5	adantr	|- ( ( ph /\ B e. ( A I C ) ) -> A e. P )
10	6	adantr	|- ( ( ph /\ B e. ( A I C ) ) -> B e. P )
11	7	adantr	|- ( ( ph /\ B e. ( A I C ) ) -> C e. P )
12		simpr	|- ( ( ph /\ B e. ( A I C ) ) -> B e. ( A I C ) )
13	1 2 3 8 9 10 11 12	tgbtwncom	|- ( ( ph /\ B e. ( A I C ) ) -> B e. ( C I A ) )
14	4	adantr	|- ( ( ph /\ B e. ( C I A ) ) -> G e. TarskiG )
15	7	adantr	|- ( ( ph /\ B e. ( C I A ) ) -> C e. P )
16	6	adantr	|- ( ( ph /\ B e. ( C I A ) ) -> B e. P )
17	5	adantr	|- ( ( ph /\ B e. ( C I A ) ) -> A e. P )
18		simpr	|- ( ( ph /\ B e. ( C I A ) ) -> B e. ( C I A ) )
19	1 2 3 14 15 16 17 18	tgbtwncom	|- ( ( ph /\ B e. ( C I A ) ) -> B e. ( A I C ) )
20	13 19	impbida	|- ( ph -> ( B e. ( A I C ) <-> B e. ( C I A ) ) )