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Theorem tgbtwncomb 25297
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 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . 4 (𝜑𝐴𝑃)
76adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . 4 (𝜑𝐵𝑃)
98adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵𝑃)
10 tgbtwncomb.3 . . . 4 (𝜑𝐶𝑃)
1110adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
12 simpr 477 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
131, 2, 3, 5, 7, 9, 11, 12tgbtwncom 25296 . 2 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐴))
144adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG)
1510adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶𝑃)
168adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵𝑃)
176adantr 481 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴𝑃)
18 simpr 477 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴))
191, 2, 3, 14, 15, 16, 17, 18tgbtwncom 25296 . 2 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶))
2013, 19impbida 876 1 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cfv 5852  (class class class)co 6610  Basecbs 15788  distcds 15878  TarskiGcstrkg 25242  Itvcitv 25248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-trkgc 25260  df-trkgb 25261  df-trkgcb 25262  df-trkg 25265
This theorem is referenced by:  colcom  25366  colrot1  25367  lnhl  25423  lncom  25430  lnrot1  25431  lnrot2  25432  mirreu3  25462
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