Metamath Proof Explorer


Theorem spthonpthon

Description: A simple path between two vertices is a path between these vertices. (Contributed by AV, 24-Jan-2021)

Ref Expression
Assertion spthonpthon F A SPathsOn G B P F A PathsOn G B P

Proof

Step Hyp Ref Expression
1 eqid Vtx G = Vtx G
2 1 spthonprop F A SPathsOn G B P G V A Vtx G B Vtx G F V P V F A TrailsOn G B P F SPaths G P
3 3simpc G V A Vtx G B Vtx G A Vtx G B Vtx G
4 3 3anim1i G V A Vtx G B Vtx G F V P V F A TrailsOn G B P F SPaths G P A Vtx G B Vtx G F V P V F A TrailsOn G B P F SPaths G P
5 spthispth F SPaths G P F Paths G P
6 5 anim2i F A TrailsOn G B P F SPaths G P F A TrailsOn G B P F Paths G P
7 6 3ad2ant3 A Vtx G B Vtx G F V P V F A TrailsOn G B P F SPaths G P F A TrailsOn G B P F Paths G P
8 1 ispthson A Vtx G B Vtx G F V P V F A PathsOn G B P F A TrailsOn G B P F Paths G P
9 8 3adant3 A Vtx G B Vtx G F V P V F A TrailsOn G B P F SPaths G P F A PathsOn G B P F A TrailsOn G B P F Paths G P
10 7 9 mpbird A Vtx G B Vtx G F V P V F A TrailsOn G B P F SPaths G P F A PathsOn G B P
11 2 4 10 3syl F A SPathsOn G B P F A PathsOn G B P