Theorem tgbtwnswapid 28955

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwnswapid.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwnswapid.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwnswapid.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnswapid.4	⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐶))
tgbtwnswapid.5	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))

Assertion

Ref	Expression
tgbtwnswapid	⊢ (𝜑 → 𝐴 = 𝐵)

Proof of Theorem tgbtwnswapid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG)
6		tgbtwnswapid.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 ∈ 𝑃)
8		simplr 781	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ 𝑃)
9		simprl 783	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐴))
10	1, 2, 3, 5, 7, 8, 9	axtgbtwnid 28928	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 = 𝑥)
11		tgbtwnswapid.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	11	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ 𝑃)
13		simprr 785	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵))
14	1, 2, 3, 5, 12, 8, 13	axtgbtwnid 28928	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥)
15	10, 14	eqtr4d 2799	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐴 = 𝐵)
16		tgbtwnswapid.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
17		tgbtwnswapid.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐶))
18		tgbtwnswapid.5	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
19	1, 2, 3, 4, 11, 6, 16, 6, 11, 17, 18	axtgpasch 28929	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐴) ∧ 𝑥 ∈ (𝐵𝐼𝐵)))
20	15, 19	r19.29a 3171	1 ⊢ (𝜑 → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  distcds 17437  TarskiGcstrkg 28889  Itvcitv 28895

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-trkgb 28911  df-trkg 28915

This theorem is used by:  legtri3  29053