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 e. _V /\ A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( TrailsOn ` G ) B ) P /\ F ( SPaths ` G ) P ) ) )
3		3simpc	|- ( ( G e. _V /\ A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) -> ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) )
4	3	3anim1i	|- ( ( ( G e. _V /\ A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A ( TrailsOn ` G ) B ) P /\ F ( SPaths ` G ) P ) ) -> ( ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _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 e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _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 e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( PathsOn ` G ) B ) P <-> ( F ( A ( TrailsOn ` G ) B ) P /\ F ( Paths ` G ) P ) ) )
9	8	3adant3	|- ( ( ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _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 e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( F e. _V /\ P e. _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 )