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Theorem tgbtwnexch2 28519
Description: Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-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)
tgbtwnexch2.1 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
tgbtwnexch2.2 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
Assertion
Ref Expression
tgbtwnexch2 (𝜑𝐶 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnexch2
StepHypRef Expression
1 simpr 484 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
2 tgbtwnexch2.1 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
32adantr 480 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
41, 3eqeltrrd 2840 . 2 ((𝜑𝐵 = 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
5 tkgeom.p . . 3 𝑃 = (Base‘𝐺)
6 tkgeom.d . . 3 = (dist‘𝐺)
7 tkgeom.i . . 3 𝐼 = (Itv‘𝐺)
8 tkgeom.g . . . 4 (𝜑𝐺 ∈ TarskiG)
98adantr 480 . . 3 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
10 tgbtwnintr.1 . . . 4 (𝜑𝐴𝑃)
1110adantr 480 . . 3 ((𝜑𝐵𝐶) → 𝐴𝑃)
12 tgbtwnintr.2 . . . 4 (𝜑𝐵𝑃)
1312adantr 480 . . 3 ((𝜑𝐵𝐶) → 𝐵𝑃)
14 tgbtwnintr.3 . . . 4 (𝜑𝐶𝑃)
1514adantr 480 . . 3 ((𝜑𝐵𝐶) → 𝐶𝑃)
16 tgbtwnintr.4 . . . 4 (𝜑𝐷𝑃)
1716adantr 480 . . 3 ((𝜑𝐵𝐶) → 𝐷𝑃)
18 simpr 484 . . 3 ((𝜑𝐵𝐶) → 𝐵𝐶)
19 tgbtwnexch2.2 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2019adantr 480 . . . . 5 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
212adantr 480 . . . . 5 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
225, 6, 7, 9, 15, 13, 11, 17, 20, 21tgbtwnintr 28516 . . . 4 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
235, 6, 7, 9, 15, 13, 11, 22tgbtwncom 28511 . . 3 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
245, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20tgbtwnouttr2 28518 . 2 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
254, 24pm2.61dane 3027 1 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476
This theorem is referenced by:  tgbtwnexch  28521  tgtrisegint  28522  tgbtwnconn1lem3  28597  legtri3  28613  miriso  28693
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