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Theorem tgbtwncomb 25184
Description: Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
tgbtwncomb.3 (𝜑𝐶𝑃)
Assertion
Ref Expression
tgbtwncomb (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))

Proof of Theorem tgbtwncomb
StepHypRef Expression
1 tkgeom.p . . 3 𝑃 = (Base‘𝐺)
2 tkgeom.d . . 3 = (dist‘𝐺)
3 tkgeom.i . . 3 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . 4 (𝜑𝐴𝑃)
76adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . 4 (𝜑𝐵𝑃)
98adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵𝑃)
10 tgbtwncomb.3 . . . 4 (𝜑𝐶𝑃)
1110adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
12 simpr 476 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
131, 2, 3, 5, 7, 9, 11, 12tgbtwncom 25183 . 2 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐴))
144adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG)
1510adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶𝑃)
168adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵𝑃)
176adantr 480 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴𝑃)
18 simpr 476 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴))
191, 2, 3, 14, 15, 16, 17, 18tgbtwncom 25183 . 2 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶))
2013, 19impbida 873 1 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152
This theorem is referenced by:  colcom  25253  colrot1  25254  lnhl  25310  lncom  25317  lnrot1  25318  lnrot2  25319  mirreu3  25349
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