Theorem tgbtwnexch3 28013

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 28007	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
11		tgbtwnexch3.6	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))
12	1, 2, 3, 4, 8, 6, 7, 11	tgbtwncom 28007	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 28012	1 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ‘cfv 6543  (class class class)co 7412  Basecbs 17149  distcds 17211  TarskiGcstrkg 27946  Itvcitv 27952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-trkgc 27967  df-trkgb 27968  df-trkgcb 27969  df-trkg 27972

This theorem is referenced by:  tgbtwnouttr2  28014  tgifscgr  28027  tgcgrxfr  28037  tgbtwnconn1lem1  28091  tgbtwnconn1lem2  28092  tgbtwnconn1lem3  28093  tgbtwnconn2  28095  tgbtwnconn3  28096  btwnhl  28133  tglineeltr  28150  miriso  28189  krippenlem  28209  outpasch  28274  hlpasch  28275