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Theorem tgbtwncomb 26274
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 483 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . 4 (𝜑𝐴𝑃)
76adantr 483 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . 4 (𝜑𝐵𝑃)
98adantr 483 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵𝑃)
10 tgbtwncomb.3 . . . 4 (𝜑𝐶𝑃)
1110adantr 483 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
12 simpr 487 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
131, 2, 3, 5, 7, 9, 11, 12tgbtwncom 26273 . 2 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐴))
144adantr 483 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG)
1510adantr 483 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶𝑃)
168adantr 483 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵𝑃)
176adantr 483 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴𝑃)
18 simpr 487 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴))
191, 2, 3, 14, 15, 16, 17, 18tgbtwncom 26273 . 2 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶))
2013, 19impbida 799 1 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  cfv 6354  (class class class)co 7155  Basecbs 16482  distcds 16573  TarskiGcstrkg 26215  Itvcitv 26221
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158  df-trkgc 26233  df-trkgb 26234  df-trkgcb 26235  df-trkg 26238
This theorem is referenced by:  colcom  26343  colrot1  26344  lnhl  26400  lncom  26407  lnrot1  26408  lnrot2  26409  mirreu3  26439
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