Theorem tgbtwnexch2 28941

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.)

Hypotheses

Ref	Expression
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 ∈ (𝐴𝐼𝐷))
tgbtwnexch2.2	⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

Assertion

Ref	Expression
tgbtwnexch2	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnexch2

Step	Hyp	Ref	Expression
1		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
2		tgbtwnexch2.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))
3	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
4	1, 3	eqeltrrd 2862	. 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
5		tkgeom.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
6		tkgeom.d	. . 3 ⊢ − = (dist‘𝐺)
7		tkgeom.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
8		tkgeom.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
10		tgbtwnintr.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃)
12		tgbtwnintr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
14		tgbtwnintr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
16		tgbtwnintr.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
17	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃)
18		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
19		tgbtwnexch2.2	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
20	19	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
21	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
22	5, 6, 7, 9, 15, 13, 11, 17, 20, 21	tgbtwnintr 28938	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
23	5, 6, 7, 9, 15, 13, 11, 22	tgbtwncom 28933	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
24	5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20	tgbtwnouttr2 28940	. 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
25	4, 24	pm2.61dane 3043	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  distcds 17417  TarskiGcstrkg 28871  Itvcitv 28877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-ov 7415  df-trkgc 28892  df-trkgb 28893  df-trkgcb 28894  df-trkg 28897

This theorem is used by:  tgbtwnexch  28943  tgtrisegint  28944  tgbtwnconn1lem3  29019  legtri3  29035  miriso  29124