Theorem tgbtwnexch3 29014

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 29008	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
11		tgbtwnexch3.6	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))
12	1, 2, 3, 4, 8, 6, 7, 11	tgbtwncom 29008	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 29013	1 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  distcds 17481  TarskiGcstrkg 28946  Itvcitv 28952

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454  df-trkgc 28967  df-trkgb 28968  df-trkgcb 28969  df-trkg 28972

This theorem is used by:  tgbtwnouttr2  29015  tgifscgr  29028  tgcgrxfr  29038  tgbtwnconn1lem1  29092  tgbtwnconn1lem2  29093  tgbtwnconn1lem3  29094  tgbtwnconn2  29096  tgbtwnconn3  29097  btwnhl  29137  tglineeltr  29156  miriso  29199  krippenlem  29219  outpasch  29290  hlpasch  29291