Theorem tgbtwnexch3 28816

Description: Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-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)
tgbtwnexch3.5	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnexch3.6	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Assertion

Ref	Expression
tgbtwnexch3	⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

Proof of Theorem tgbtwnexch3

Step	Hyp	Ref	Expression
1		tkgeom.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		tkgeom.d	. 2 ⊢ − = (dist‘𝐺)
3		tkgeom.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		tkgeom.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgbtwnintr.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
6		tgbtwnintr.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7		tgbtwnintr.4	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
8		tgbtwnintr.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9		tgbtwnexch3.5	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
10	1, 2, 3, 4, 8, 5, 6, 9	tgbtwncom 28810	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
11		tgbtwnexch3.6	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))
12	1, 2, 3, 4, 8, 6, 7, 11	tgbtwncom 28810	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 28815	1 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  distcds 17343  TarskiGcstrkg 28749  Itvcitv 28755

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 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

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 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-trkgc 28770  df-trkgb 28771  df-trkgcb 28772  df-trkg 28775

This theorem is used by:  tgbtwnouttr2  28817  tgifscgr  28830  tgcgrxfr  28840  tgbtwnconn1lem1  28894  tgbtwnconn1lem2  28895  tgbtwnconn1lem3  28896  tgbtwnconn2  28898  tgbtwnconn3  28899  btwnhl  28939  tglineeltr  28957  miriso  29000  krippenlem  29020  outpasch  29090  hlpasch  29091