Theorem tgbtwnexch2 28216

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 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
2		tgbtwnexch2.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))
3	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
4	1, 3	eqeltrrd 2826	. 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 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
10		tgbtwnintr.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃)
12		tgbtwnintr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
14		tgbtwnintr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
16		tgbtwnintr.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃)
18		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
19		tgbtwnexch2.2	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
21	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
22	5, 6, 7, 9, 15, 13, 11, 17, 20, 21	tgbtwnintr 28213	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
23	5, 6, 7, 9, 15, 13, 11, 22	tgbtwncom 28208	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
24	5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20	tgbtwnouttr2 28215	. 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
25	4, 24	pm2.61dane 3021	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ‘cfv 6533  (class class class)co 7401  Basecbs 17143  distcds 17205  TarskiGcstrkg 28147  Itvcitv 28153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-trkgc 28168  df-trkgb 28169  df-trkgcb 28170  df-trkg 28173

This theorem is referenced by:  tgbtwnexch  28218  tgtrisegint  28219  tgbtwnconn1lem3  28294  legtri3  28310  miriso  28390