tgbtwncomb

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

	Ref	Expression
Hypotheses	tkgeom.p	$⊢ P = {Base}_{G}$
	tkgeom.d	$⊢ \underset{˙}{-} = dist (G)$
	tkgeom.i	$⊢ I = Itv (G)$
	tkgeom.g	$⊢ φ \to G \in 𝒢_{Tarski}$
	tgbtwntriv2.1	$⊢ φ \to A \in P$
	tgbtwntriv2.2	$⊢ φ \to B \in P$
	tgbtwncomb.3	$⊢ φ \to C \in P$
Assertion	tgbtwncomb	$⊢ φ \to (B \in (A I C) \leftrightarrow B \in (C I A))$

Step	Hyp	Ref	Expression
1		tkgeom.p	$⊢ P = {Base}_{G}$
2		tkgeom.d	$⊢ \underset{˙}{-} = dist (G)$
3		tkgeom.i	$⊢ I = Itv (G)$
4		tkgeom.g	$⊢ φ \to G \in 𝒢_{Tarski}$
5		tgbtwntriv2.1	$⊢ φ \to A \in P$
6		tgbtwntriv2.2	$⊢ φ \to B \in P$
7		tgbtwncomb.3	$⊢ φ \to C \in P$
8	4	adantr	$⊢ (φ \land B \in (A I C)) \to G \in 𝒢_{Tarski}$
9	5	adantr	$⊢ (φ \land B \in (A I C)) \to A \in P$
10	6	adantr	$⊢ (φ \land B \in (A I C)) \to B \in P$
11	7	adantr	$⊢ (φ \land B \in (A I C)) \to C \in P$
12		simpr	$⊢ (φ \land B \in (A I C)) \to B \in (A I C)$
13	1 2 3 8 9 10 11 12	tgbtwncom	$⊢ (φ \land B \in (A I C)) \to B \in (C I A)$
14	4	adantr	$⊢ (φ \land B \in (C I A)) \to G \in 𝒢_{Tarski}$
15	7	adantr	$⊢ (φ \land B \in (C I A)) \to C \in P$
16	6	adantr	$⊢ (φ \land B \in (C I A)) \to B \in P$
17	5	adantr	$⊢ (φ \land B \in (C I A)) \to A \in P$
18		simpr	$⊢ (φ \land B \in (C I A)) \to B \in (C I A)$
19	1 2 3 14 15 16 17 18	tgbtwncom	$⊢ (φ \land B \in (C I A)) \to B \in (A I C)$
20	13 19	impbida	$⊢ φ \to (B \in (A I C) \leftrightarrow B \in (C I A))$

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