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Theorem tgbtwnexch2 28423
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 483 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
2 tgbtwnexch2.1 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
32adantr 479 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
41, 3eqeltrrd 2827 . 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 479 . . 3 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
10 tgbtwnintr.1 . . . 4 (𝜑𝐴𝑃)
1110adantr 479 . . 3 ((𝜑𝐵𝐶) → 𝐴𝑃)
12 tgbtwnintr.2 . . . 4 (𝜑𝐵𝑃)
1312adantr 479 . . 3 ((𝜑𝐵𝐶) → 𝐵𝑃)
14 tgbtwnintr.3 . . . 4 (𝜑𝐶𝑃)
1514adantr 479 . . 3 ((𝜑𝐵𝐶) → 𝐶𝑃)
16 tgbtwnintr.4 . . . 4 (𝜑𝐷𝑃)
1716adantr 479 . . 3 ((𝜑𝐵𝐶) → 𝐷𝑃)
18 simpr 483 . . 3 ((𝜑𝐵𝐶) → 𝐵𝐶)
19 tgbtwnexch2.2 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2019adantr 479 . . . . 5 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
212adantr 479 . . . . 5 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
225, 6, 7, 9, 15, 13, 11, 17, 20, 21tgbtwnintr 28420 . . . 4 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
235, 6, 7, 9, 15, 13, 11, 22tgbtwncom 28415 . . 3 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
245, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20tgbtwnouttr2 28422 . 2 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
254, 24pm2.61dane 3019 1 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wne 2930  cfv 6554  (class class class)co 7424  Basecbs 17213  distcds 17275  TarskiGcstrkg 28354  Itvcitv 28360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562  df-ov 7427  df-trkgc 28375  df-trkgb 28376  df-trkgcb 28377  df-trkg 28380
This theorem is referenced by:  tgbtwnexch  28425  tgtrisegint  28426  tgbtwnconn1lem3  28501  legtri3  28517  miriso  28597
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