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Theorem tgbtwnswapid 28515
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 726 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG)
6 tgbtwnswapid.1 . . . . 5 (𝜑𝐴𝑃)
76ad2antrr 726 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴𝑃)
8 simplr 769 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥𝑃)
9 simprl 771 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐴))
101, 2, 3, 5, 7, 8, 9axtgbtwnid 28489 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 = 𝑥)
11 tgbtwnswapid.2 . . . . 5 (𝜑𝐵𝑃)
1211ad2antrr 726 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵𝑃)
13 simprr 773 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵))
141, 2, 3, 5, 12, 8, 13axtgbtwnid 28489 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥)
1510, 14eqtr4d 2778 . 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 28490 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵)))
2015, 19r19.29a 3160 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-trkgb 28472  df-trkg 28476
This theorem is referenced by:  legtri3  28613
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