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 ) |