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Theorem tgbtwnswapid 25187
 Description: If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 16-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwnswapid.1 (𝜑𝐴𝑃)
tgbtwnswapid.2 (𝜑𝐵𝑃)
tgbtwnswapid.3 (𝜑𝐶𝑃)
tgbtwnswapid.4 (𝜑𝐴 ∈ (𝐵𝐼𝐶))
tgbtwnswapid.5 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
Assertion
Ref Expression
tgbtwnswapid (𝜑𝐴 = 𝐵)

Proof of Theorem tgbtwnswapid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . 4 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . 4 = (dist‘𝐺)
3 tkgeom.i . . . 4 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 758 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG)
6 tgbtwnswapid.1 . . . . 5 (𝜑𝐴𝑃)
76ad2antrr 758 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴𝑃)
8 simplr 788 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥𝑃)
9 simprl 790 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐴))
101, 2, 3, 5, 7, 8, 9axtgbtwnid 25165 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 = 𝑥)
11 tgbtwnswapid.2 . . . . 5 (𝜑𝐵𝑃)
1211ad2antrr 758 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵𝑃)
13 simprr 792 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵))
141, 2, 3, 5, 12, 8, 13axtgbtwnid 25165 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥)
1510, 14eqtr4d 2647 . 2 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 = 𝐵)
16 tgbtwnswapid.3 . . 3 (𝜑𝐶𝑃)
17 tgbtwnswapid.4 . . 3 (𝜑𝐴 ∈ (𝐵𝐼𝐶))
18 tgbtwnswapid.5 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
191, 2, 3, 4, 11, 6, 16, 6, 11, 17, 18axtgpasch 25166 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵)))
2015, 19r19.29a 3060 1 (𝜑𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ 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-trkgb 25148  df-trkg 25152 This theorem is referenced by:  legtri3  25285
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