Theorem tgbtwnexch2 28746

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 489	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
2		tgbtwnexch2.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))
3	2	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
4	1, 3	eqeltrrd 2864	. 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 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐺 ∈ TarskiG)
10		tgbtwnintr.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ∈ 𝑃)
12		tgbtwnintr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ 𝑃)
14		tgbtwnintr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15	14	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ 𝑃)
16		tgbtwnintr.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
17	16	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐷 ∈ 𝑃)
18		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
19		tgbtwnexch2.2	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
20	19	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
21	2	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
22	5, 6, 7, 9, 15, 13, 11, 17, 20, 21	tgbtwnintr 28743	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
23	5, 6, 7, 9, 15, 13, 11, 22	tgbtwncom 28738	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
24	5, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20	tgbtwnouttr2 28745	. 2 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
25	4, 24	pm2.61dane 3045	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  distcds 17320  TarskiGcstrkg 28677  Itvcitv 28683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-trkgc 28698  df-trkgb 28699  df-trkgcb 28700  df-trkg 28703

This theorem is referenced by:  tgbtwnexch  28748  tgtrisegint  28749  tgbtwnconn1lem3  28824  legtri3  28840  miriso  28928