| Step |
Hyp |
Ref |
Expression |
| 1 |
|
2wlkd.p |
|- P = <" A B C "> |
| 2 |
|
2wlkd.f |
|- F = <" J K "> |
| 3 |
|
2wlkd.s |
|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) ) |
| 4 |
|
2wlkd.n |
|- ( ph -> ( A =/= B /\ B =/= C ) ) |
| 5 |
|
2wlkd.e |
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) ) |
| 6 |
1 2 3 4 5
|
2wlkdlem7 |
|- ( ph -> ( J e. _V /\ K e. _V ) ) |
| 7 |
|
s2fv0 |
|- ( J e. _V -> ( <" J K "> ` 0 ) = J ) |
| 8 |
|
s2fv1 |
|- ( K e. _V -> ( <" J K "> ` 1 ) = K ) |
| 9 |
7 8
|
anim12i |
|- ( ( J e. _V /\ K e. _V ) -> ( ( <" J K "> ` 0 ) = J /\ ( <" J K "> ` 1 ) = K ) ) |
| 10 |
6 9
|
syl |
|- ( ph -> ( ( <" J K "> ` 0 ) = J /\ ( <" J K "> ` 1 ) = K ) ) |
| 11 |
2
|
fveq1i |
|- ( F ` 0 ) = ( <" J K "> ` 0 ) |
| 12 |
11
|
eqeq1i |
|- ( ( F ` 0 ) = J <-> ( <" J K "> ` 0 ) = J ) |
| 13 |
2
|
fveq1i |
|- ( F ` 1 ) = ( <" J K "> ` 1 ) |
| 14 |
13
|
eqeq1i |
|- ( ( F ` 1 ) = K <-> ( <" J K "> ` 1 ) = K ) |
| 15 |
12 14
|
anbi12i |
|- ( ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) <-> ( ( <" J K "> ` 0 ) = J /\ ( <" J K "> ` 1 ) = K ) ) |
| 16 |
10 15
|
sylibr |
|- ( ph -> ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) ) |