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Theorem tgbtwnexch3 26292
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 (𝜑𝐺 ∈ TarskiG)
tgbtwnintr.1 (𝜑𝐴𝑃)
tgbtwnintr.2 (𝜑𝐵𝑃)
tgbtwnintr.3 (𝜑𝐶𝑃)
tgbtwnintr.4 (𝜑𝐷𝑃)
tgbtwnexch3.5 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnexch3.6 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Assertion
Ref Expression
tgbtwnexch3 (𝜑𝐶 ∈ (𝐵𝐼𝐷))

Proof of Theorem tgbtwnexch3
StepHypRef 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 (𝜑𝐺 ∈ TarskiG)
5 tgbtwnintr.2 . 2 (𝜑𝐵𝑃)
6 tgbtwnintr.3 . 2 (𝜑𝐶𝑃)
7 tgbtwnintr.4 . 2 (𝜑𝐷𝑃)
8 tgbtwnintr.1 . 2 (𝜑𝐴𝑃)
9 tgbtwnexch3.5 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
101, 2, 3, 4, 8, 5, 6, 9tgbtwncom 26286 . 2 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
11 tgbtwnexch3.6 . . 3 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
121, 2, 3, 4, 8, 6, 7, 11tgbtwncom 26286 . 2 (𝜑𝐶 ∈ (𝐷𝐼𝐴))
131, 2, 3, 4, 5, 6, 7, 8, 10, 12tgbtwnintr 26291 1 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cfv 6328  (class class class)co 7139  Basecbs 16479  distcds 16570  TarskiGcstrkg 26228  Itvcitv 26234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-trkgc 26246  df-trkgb 26247  df-trkgcb 26248  df-trkg 26251
This theorem is referenced by:  tgbtwnouttr2  26293  tgifscgr  26306  tgcgrxfr  26316  tgbtwnconn1lem1  26370  tgbtwnconn1lem2  26371  tgbtwnconn1lem3  26372  tgbtwnconn2  26374  tgbtwnconn3  26375  btwnhl  26412  tglineeltr  26429  miriso  26468  krippenlem  26488  outpasch  26553  hlpasch  26554
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