Metamath Proof Explorer


Theorem tgbtwnexch3

Description: Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019)

Ref Expression
Hypotheses tkgeom.p P=BaseG
tkgeom.d -˙=distG
tkgeom.i I=ItvG
tkgeom.g φG𝒢Tarski
tgbtwnintr.1 φAP
tgbtwnintr.2 φBP
tgbtwnintr.3 φCP
tgbtwnintr.4 φDP
tgbtwnexch3.5 φBAIC
tgbtwnexch3.6 φCAID
Assertion tgbtwnexch3 φCBID

Proof

Step Hyp Ref Expression
1 tkgeom.p P=BaseG
2 tkgeom.d -˙=distG
3 tkgeom.i I=ItvG
4 tkgeom.g φG𝒢Tarski
5 tgbtwnintr.1 φAP
6 tgbtwnintr.2 φBP
7 tgbtwnintr.3 φCP
8 tgbtwnintr.4 φDP
9 tgbtwnexch3.5 φBAIC
10 tgbtwnexch3.6 φCAID
11 1 2 3 4 5 6 7 9 tgbtwncom φBCIA
12 1 2 3 4 5 7 8 10 tgbtwncom φCDIA
13 1 2 3 4 6 7 8 5 11 12 tgbtwnintr φCBID