Theorem tgbtwnexch3 28663

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 28657	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
11		tgbtwnexch3.6	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))
12	1, 2, 3, 4, 8, 6, 7, 11	tgbtwncom 28657	. 2 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
13	1, 2, 3, 4, 5, 6, 7, 8, 10, 12	tgbtwnintr 28662	1 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  distcds 17295  TarskiGcstrkg 28596  Itvcitv 28602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5256

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399  df-trkgc 28617  df-trkgb 28618  df-trkgcb 28619  df-trkg 28622

This theorem is referenced by:  tgbtwnouttr2  28664  tgifscgr  28677  tgcgrxfr  28687  tgbtwnconn1lem1  28741  tgbtwnconn1lem2  28742  tgbtwnconn1lem3  28743  tgbtwnconn2  28745  tgbtwnconn3  28746  btwnhl  28783  tglineeltr  28800  miriso  28843  krippenlem  28863  outpasch  28928  hlpasch  28929