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	$⊢ \underset{˙}{-} = dist (G)$
	tkgeom.i	$⊢ I = Itv (G)$
	tkgeom.g	$⊢ φ \to G \in 𝒢_{Tarski}$
	tgbtwnswapid.1	$⊢ φ \to A \in P$
	tgbtwnswapid.2	$⊢ φ \to B \in P$
	tgbtwnswapid.3	$⊢ φ \to C \in P$
	tgbtwnswapid.4	$⊢ φ \to A \in (B I C)$
	tgbtwnswapid.5	$⊢ φ \to B \in (A I C)$
Assertion	tgbtwnswapid	$⊢ φ \to A = B$

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	$⊢ P = {Base}_{G}$
2		tkgeom.d	$⊢ \underset{˙}{-} = dist (G)$
3		tkgeom.i	$⊢ I = Itv (G)$
4		tkgeom.g	$⊢ φ \to G \in 𝒢_{Tarski}$
5		tgbtwnswapid.1	$⊢ φ \to A \in P$
6		tgbtwnswapid.2	$⊢ φ \to B \in P$
7		tgbtwnswapid.3	$⊢ φ \to C \in P$
8		tgbtwnswapid.4	$⊢ φ \to A \in (B I C)$
9		tgbtwnswapid.5	$⊢ φ \to B \in (A I C)$
10	4	ad2antrr	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to G \in 𝒢_{Tarski}$
11	5	ad2antrr	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to A \in P$
12		simplr	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to x \in P$
13		simprl	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to x \in (A I A)$
14	1 2 3 10 11 12 13	axtgbtwnid	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to A = x$
15	6	ad2antrr	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to B \in P$
16		simprr	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to x \in (B I B)$
17	1 2 3 10 15 12 16	axtgbtwnid	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to B = x$
18	14 17	eqtr4d	$⊢ ((φ \land x \in P) \land (x \in (A I A) \land x \in (B I B))) \to A = B$
19	1 2 3 4 6 5 7 5 6 8 9	axtgpasch	$⊢ φ \to \exists x \in P (x \in (A I A) \land x \in (B I B))$
20	18 19	r19.29a	$⊢ φ \to A = B$

Metamath Proof Explorer

Theorem tgbtwnswapid

Proof