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Theorem tgbtwncomb 28008
Description: Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Baseβ€˜πΊ)
tkgeom.d βˆ’ = (distβ€˜πΊ)
tkgeom.i 𝐼 = (Itvβ€˜πΊ)
tkgeom.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgbtwntriv2.1 (πœ‘ β†’ 𝐴 ∈ 𝑃)
tgbtwntriv2.2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
tgbtwncomb.3 (πœ‘ β†’ 𝐢 ∈ 𝑃)
Assertion
Ref Expression
tgbtwncomb (πœ‘ β†’ (𝐡 ∈ (𝐴𝐼𝐢) ↔ 𝐡 ∈ (𝐢𝐼𝐴)))

Proof of Theorem tgbtwncomb
StepHypRef Expression
1 tkgeom.p . . 3 𝑃 = (Baseβ€˜πΊ)
2 tkgeom.d . . 3 βˆ’ = (distβ€˜πΊ)
3 tkgeom.i . . 3 𝐼 = (Itvβ€˜πΊ)
4 tkgeom.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54adantr 480 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐴𝐼𝐢)) β†’ 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑃)
76adantr 480 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐴𝐼𝐢)) β†’ 𝐴 ∈ 𝑃)
8 tgbtwntriv2.2 . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑃)
98adantr 480 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐴𝐼𝐢)) β†’ 𝐡 ∈ 𝑃)
10 tgbtwncomb.3 . . . 4 (πœ‘ β†’ 𝐢 ∈ 𝑃)
1110adantr 480 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐴𝐼𝐢)) β†’ 𝐢 ∈ 𝑃)
12 simpr 484 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐴𝐼𝐢)) β†’ 𝐡 ∈ (𝐴𝐼𝐢))
131, 2, 3, 5, 7, 9, 11, 12tgbtwncom 28007 . 2 ((πœ‘ ∧ 𝐡 ∈ (𝐴𝐼𝐢)) β†’ 𝐡 ∈ (𝐢𝐼𝐴))
144adantr 480 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐢𝐼𝐴)) β†’ 𝐺 ∈ TarskiG)
1510adantr 480 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐢𝐼𝐴)) β†’ 𝐢 ∈ 𝑃)
168adantr 480 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐢𝐼𝐴)) β†’ 𝐡 ∈ 𝑃)
176adantr 480 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐢𝐼𝐴)) β†’ 𝐴 ∈ 𝑃)
18 simpr 484 . . 3 ((πœ‘ ∧ 𝐡 ∈ (𝐢𝐼𝐴)) β†’ 𝐡 ∈ (𝐢𝐼𝐴))
191, 2, 3, 14, 15, 16, 17, 18tgbtwncom 28007 . 2 ((πœ‘ ∧ 𝐡 ∈ (𝐢𝐼𝐴)) β†’ 𝐡 ∈ (𝐴𝐼𝐢))
2013, 19impbida 798 1 (πœ‘ β†’ (𝐡 ∈ (𝐴𝐼𝐢) ↔ 𝐡 ∈ (𝐢𝐼𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  distcds 17211  TarskiGcstrkg 27946  Itvcitv 27952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-trkgc 27967  df-trkgb 27968  df-trkgcb 27969  df-trkg 27972
This theorem is referenced by:  colcom  28077  colrot1  28078  lnhl  28134  lncom  28141  lnrot1  28142  lnrot2  28143  mirreu3  28173
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