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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwntriv2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwntriv2.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgbtwncomb.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
Assertion	tgbtwncomb	⊢ ( 𝜑 → ( 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ↔ 𝐵 ∈ ( 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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgbtwntriv2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgbtwntriv2.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgbtwncomb.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8	4	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐺 ∈ TarskiG )
9	5	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐴 ∈ 𝑃 )
10	6	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐵 ∈ 𝑃 )
11	7	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐶 ∈ 𝑃 )
12		simpr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
13	1 2 3 8 9 10 11 12	tgbtwncom	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) )
14	4	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐺 ∈ TarskiG )
15	7	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐶 ∈ 𝑃 )
16	6	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐵 ∈ 𝑃 )
17	5	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐴 ∈ 𝑃 )
18		simpr	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) )
19	1 2 3 14 15 16 17 18	tgbtwncom	⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
20	13 19	impbida	⊢ ( 𝜑 → ( 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ↔ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) )