Theorem tgbtwnswapid

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)

	Ref	Expression
Hypotheses	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 ∈ ( 𝐵 𝐼 𝐶 ) )
	tgbtwnswapid.5	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
Assertion	tgbtwnswapid	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tkgeom.d	⊢ − = ( dist ‘ 𝐺 )
3		tkgeom.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tkgeom.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgbtwnswapid.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgbtwnswapid.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgbtwnswapid.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgbtwnswapid.4	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐶 ) )
9		tgbtwnswapid.5	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
10	4	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐺 ∈ TarskiG )
11	5	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐴 ∈ 𝑃 )
12		simplr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝑥 ∈ 𝑃 )
13		simprl	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) )
14	1 2 3 10 11 12 13	axtgbtwnid	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐴 = 𝑥 )
15	6	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐵 ∈ 𝑃 )
16		simprr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) )
17	1 2 3 10 15 12 16	axtgbtwnid	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐵 = 𝑥 )
18	14 17	eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐴 = 𝐵 )
19	1 2 3 4 6 5 7 5 6 8 9	axtgpasch	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑃 ( 𝑥 ∈ ( 𝐴 𝐼 𝐴 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) )
20	18 19	r19.29a	⊢ ( 𝜑 → 𝐴 = 𝐵 )