Theorem tgbtwnexch3 28522

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 28516	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
11		tgbtwnexch3.6	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))
12	1, 2, 3, 4, 8, 6, 7, 11	tgbtwncom 28516	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 28521	1 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ‘cfv 6575  (class class class)co 7450  Basecbs 17260  distcds 17322  TarskiGcstrkg 28455  Itvcitv 28461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-ov 7453  df-trkgc 28476  df-trkgb 28477  df-trkgcb 28478  df-trkg 28481

This theorem is referenced by:  tgbtwnouttr2  28523  tgifscgr  28536  tgcgrxfr  28546  tgbtwnconn1lem1  28600  tgbtwnconn1lem2  28601  tgbtwnconn1lem3  28602  tgbtwnconn2  28604  tgbtwnconn3  28605  btwnhl  28642  tglineeltr  28659  miriso  28698  krippenlem  28718  outpasch  28783  hlpasch  28784