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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwnintr.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwnintr.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgbtwnintr.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgbtwnintr.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	tgbtwnexch2.1	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
	tgbtwnexch2.2	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
Assertion	tgbtwnexch2	⊢ ( 𝜑 → 𝐶 ∈ ( 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	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgbtwnintr.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgbtwnintr.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgbtwnintr.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgbtwnintr.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
9		tgbtwnexch2.1	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
10		tgbtwnexch2.2	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
11		simpr	⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐵 = 𝐶 )
12	9	adantr	⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
13	11 12	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝐵 = 𝐶 ) → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) )
14	4	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐺 ∈ TarskiG )
15	5	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐴 ∈ 𝑃 )
16	6	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ∈ 𝑃 )
17	7	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐶 ∈ 𝑃 )
18	8	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐷 ∈ 𝑃 )
19		simpr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ≠ 𝐶 )
20	10	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
21	9	adantr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
22	1 2 3 14 17 16 15 18 20 21	tgbtwnintr	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) )
23	1 2 3 14 17 16 15 22	tgbtwncom	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
24	1 2 3 14 15 16 17 18 19 23 20	tgbtwnouttr2	⊢ ( ( 𝜑 ∧ 𝐵 ≠ 𝐶 ) → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) )
25	13 24	pm2.61dane	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) )