Metamath Proof Explorer

Theorem btwnswapid2

Description: If you can swap arguments one and three of a betweenness statement, then those arguments are identical. (Contributed by Scott Fenton, 7-Oct-2013)

Ref Expression
Assertion btwnswapid2 
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. /\ C Btwn <. B , A >. ) -> A = C ) )

Proof

Step Hyp Ref Expression
1 btwncom 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Btwn <. B , C >. <-> A Btwn <. C , B >. ) )
2 3anrev 
 |-  ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( C e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) )
3 btwncom 
 |-  ( ( N e. NN /\ ( C e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( C Btwn <. B , A >. <-> C Btwn <. A , B >. ) )
4 2 3 sylan2b 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. B , A >. <-> C Btwn <. A , B >. ) )
5 1 4 anbi12d 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. /\ C Btwn <. B , A >. ) <-> ( A Btwn <. C , B >. /\ C Btwn <. A , B >. ) ) )
6 3ancomb 
 |-  ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
7 btwnswapid 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( A Btwn <. C , B >. /\ C Btwn <. A , B >. ) -> A = C ) )
8 6 7 sylan2b 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. C , B >. /\ C Btwn <. A , B >. ) -> A = C ) )
9 5 8 sylbid 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. /\ C Btwn <. B , A >. ) -> A = C ) )