Theorem tgbtwnexch2 26290

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 488	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
2		tgbtwnexch2.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))
3	2	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
4	1, 3	eqeltrrd 2891	. 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 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
10		tgbtwnintr.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	10	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃)
12		tgbtwnintr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
14		tgbtwnintr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
16		tgbtwnintr.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
17	16	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃)
18		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
19		tgbtwnexch2.2	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
20	19	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
21	2	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
22	5, 6, 7, 9, 15, 13, 11, 17, 20, 21	tgbtwnintr 26287	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
23	5, 6, 7, 9, 15, 13, 11, 22	tgbtwncom 26282	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
24	5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20	tgbtwnouttr2 26289	. 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
25	4, 24	pm2.61dane 3074	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  distcds 16566  TarskiGcstrkg 26224  Itvcitv 26230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-trkgc 26242  df-trkgb 26243  df-trkgcb 26244  df-trkg 26247

This theorem is referenced by:  tgbtwnexch  26292  tgtrisegint  26293  tgbtwnconn1lem3  26368  legtri3  26384  miriso  26464