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Theorem tgbtwncomb 26382
 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 484 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . 4 (𝜑𝐴𝑃)
76adantr 484 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . 4 (𝜑𝐵𝑃)
98adantr 484 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵𝑃)
10 tgbtwncomb.3 . . . 4 (𝜑𝐶𝑃)
1110adantr 484 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
12 simpr 488 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
131, 2, 3, 5, 7, 9, 11, 12tgbtwncom 26381 . 2 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐴))
144adantr 484 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG)
1510adantr 484 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶𝑃)
168adantr 484 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵𝑃)
176adantr 484 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴𝑃)
18 simpr 488 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴))
191, 2, 3, 14, 15, 16, 17, 18tgbtwncom 26381 . 2 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶))
2013, 19impbida 800 1 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  distcds 16632  TarskiGcstrkg 26323  Itvcitv 26329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-ov 7153  df-trkgc 26341  df-trkgb 26342  df-trkgcb 26343  df-trkg 26346 This theorem is referenced by:  colcom  26451  colrot1  26452  lnhl  26508  lncom  26515  lnrot1  26516  lnrot2  26517  mirreu3  26547
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