Metamath Proof Explorer


Theorem 3spthond

Description: A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021) (Revised by AV, 24-Mar-2021)

Ref Expression
Hypotheses 3wlkd.p P = ⟨“ ABCD ”⟩
3wlkd.f F = ⟨“ JKL ”⟩
3wlkd.s φ A V B V C V D V
3wlkd.n φ A B A C B C B D C D
3wlkd.e φ A B I J B C I K C D I L
3wlkd.v V = Vtx G
3wlkd.i I = iEdg G
3trld.n φ J K J L K L
3spthd.n φ A D
Assertion 3spthond φ F A SPathsOn G D P

Proof

Step Hyp Ref Expression
1 3wlkd.p P = ⟨“ ABCD ”⟩
2 3wlkd.f F = ⟨“ JKL ”⟩
3 3wlkd.s φ A V B V C V D V
4 3wlkd.n φ A B A C B C B D C D
5 3wlkd.e φ A B I J B C I K C D I L
6 3wlkd.v V = Vtx G
7 3wlkd.i I = iEdg G
8 3trld.n φ J K J L K L
9 3spthd.n φ A D
10 1 2 3 4 5 6 7 8 3trlond φ F A TrailsOn G D P
11 1 2 3 4 5 6 7 8 9 3spthd φ F SPaths G P
12 3 simplld φ A V
13 3 simprrd φ D V
14 s3cli ⟨“ JKL ”⟩ Word V
15 2 14 eqeltri F Word V
16 s4cli ⟨“ ABCD ”⟩ Word V
17 1 16 eqeltri P Word V
18 15 17 pm3.2i F Word V P Word V
19 18 a1i φ F Word V P Word V
20 6 isspthson A V D V F Word V P Word V F A SPathsOn G D P F A TrailsOn G D P F SPaths G P
21 12 13 19 20 syl21anc φ F A SPathsOn G D P F A TrailsOn G D P F SPaths G P
22 10 11 21 mpbir2and φ F A SPathsOn G D P