Metamath Proof Explorer

Theorem usgr2wlkspth

Description: In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018) (Revised by AV, 27-Jan-2021) (Proof shortened by AV, 31-Oct-2021)

Ref Expression
Assertion usgr2wlkspth ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 ↔ 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ) )

Proof

Step Hyp Ref Expression
1 simpl31 ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
2 simp2 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( 𝑃 ‘ 0 ) = 𝐴 )
3 simp3 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 )
4 2 3 neeq12d ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ↔ 𝐴 ≠ 𝐵 ) )
5 4 bicomd ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
6 5 3anbi3d ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ↔ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) )
7 usgr2wlkspthlem1 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → Fun ◡ 𝐹 )
8 7 ex ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → Fun ◡ 𝐹 ) )
9 8 3ad2ant1 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → Fun ◡ 𝐹 ) )
10 6 9 sylbid ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) → Fun ◡ 𝐹 ) )
11 10 3ad2ant3 ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) → Fun ◡ 𝐹 ) )
12 11 imp ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → Fun ◡ 𝐹 )
13 istrl ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
14 1 12 13 sylanbrc ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
15 usgr2wlkspthlem2 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → Fun ◡ 𝑃 )
16 15 ex ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → Fun ◡ 𝑃 ) )
17 16 3ad2ant1 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → Fun ◡ 𝑃 ) )
18 6 17 sylbid ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) → Fun ◡ 𝑃 ) )
19 18 3ad2ant3 ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) → Fun ◡ 𝑃 ) )
20 19 imp ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → Fun ◡ 𝑃 )
21 isspth ⊢ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝑃 ) )
22 14 20 21 sylanbrc ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 )
23 3simpc ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) )
24 23 3ad2ant3 ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) )
25 24 adantr ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) )
26 3anass ⊢ ( ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ↔ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
27 22 25 26 sylanbrc ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) )
28 3simpa ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) )
29 28 adantr ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) )
30 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
31 30 isspthonpth ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) → ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ↔ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
32 29 31 syl ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ↔ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
33 27 32 mpbird ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) ) → 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
34 33 ex ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) → 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ) )
35 30 wlkonprop ⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
36 3simpc ⊢ ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) )
37 36 3anim1i ⊢ ( ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
38 35 37 syl ⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
39 34 38 syl11 ⊢ ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ) )
40 spthonpthon ⊢ ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
41 pthontrlon ⊢ ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
42 trlsonwlkon ⊢ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
43 40 41 42 3syl ⊢ ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
44 39 43 impbid1 ⊢ ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 ↔ 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ) )