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Theorem tgbtwnexch3 25610
 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 25604 . 2 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
11 tgbtwnexch3.6 . . 3 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
121, 2, 3, 4, 8, 6, 7, 11tgbtwncom 25604 . 2 (𝜑𝐶 ∈ (𝐷𝐼𝐴))
131, 2, 3, 4, 5, 6, 7, 8, 10, 12tgbtwnintr 25609 1 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573 This theorem is referenced by:  tgbtwnouttr2  25611  tgifscgr  25624  tgcgrxfr  25634  tgbtwnconn1lem1  25688  tgbtwnconn1lem2  25689  tgbtwnconn1lem3  25690  tgbtwnconn2  25692  tgbtwnconn3  25693  btwnhl  25730  tglineeltr  25747  miriso  25786  krippenlem  25806  outpasch  25868  hlpasch  25869
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