| 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
|
2wlkdlem8 |
|- ( ph -> ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) ) |
| 7 |
|
fveq2 |
|- ( ( F ` 0 ) = J -> ( I ` ( F ` 0 ) ) = ( I ` J ) ) |
| 8 |
7
|
adantr |
|- ( ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) -> ( I ` ( F ` 0 ) ) = ( I ` J ) ) |
| 9 |
8
|
sseq2d |
|- ( ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) -> ( { A , B } C_ ( I ` ( F ` 0 ) ) <-> { A , B } C_ ( I ` J ) ) ) |
| 10 |
|
fveq2 |
|- ( ( F ` 1 ) = K -> ( I ` ( F ` 1 ) ) = ( I ` K ) ) |
| 11 |
10
|
adantl |
|- ( ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) -> ( I ` ( F ` 1 ) ) = ( I ` K ) ) |
| 12 |
11
|
sseq2d |
|- ( ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) -> ( { B , C } C_ ( I ` ( F ` 1 ) ) <-> { B , C } C_ ( I ` K ) ) ) |
| 13 |
9 12
|
anbi12d |
|- ( ( ( F ` 0 ) = J /\ ( F ` 1 ) = K ) -> ( ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) ) <-> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) ) ) |
| 14 |
6 13
|
syl |
|- ( ph -> ( ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) ) <-> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) ) ) |
| 15 |
5 14
|
mpbird |
|- ( ph -> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) ) ) |