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	|- ( ph -> G e. TarskiG )
	tgbtwnswapid.1	|- ( ph -> A e. P )
	tgbtwnswapid.2	|- ( ph -> B e. P )
	tgbtwnswapid.3	|- ( ph -> C e. P )
	tgbtwnswapid.4	|- ( ph -> A e. ( B I C ) )
	tgbtwnswapid.5	|- ( ph -> B e. ( A I C ) )
Assertion	tgbtwnswapid	|- ( ph -> 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	|- ( ph -> G e. TarskiG )
5		tgbtwnswapid.1	|- ( ph -> A e. P )
6		tgbtwnswapid.2	|- ( ph -> B e. P )
7		tgbtwnswapid.3	|- ( ph -> C e. P )
8		tgbtwnswapid.4	|- ( ph -> A e. ( B I C ) )
9		tgbtwnswapid.5	|- ( ph -> B e. ( A I C ) )
10	4	ad2antrr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> G e. TarskiG )
11	5	ad2antrr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> A e. P )
12		simplr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> x e. P )
13		simprl	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> x e. ( A I A ) )
14	1 2 3 10 11 12 13	axtgbtwnid	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> A = x )
15	6	ad2antrr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> B e. P )
16		simprr	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> x e. ( B I B ) )
17	1 2 3 10 15 12 16	axtgbtwnid	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> B = x )
18	14 17	eqtr4d	|- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I A ) /\ x e. ( B I B ) ) ) -> A = B )
19	1 2 3 4 6 5 7 5 6 8 9	axtgpasch	|- ( ph -> E. x e. P ( x e. ( A I A ) /\ x e. ( B I B ) ) )
20	18 19	r19.29a	|- ( ph -> A = B )