Step |
Hyp |
Ref |
Expression |
1 |
|
choccl |
|- ( B e. CH -> ( _|_ ` B ) e. CH ) |
2 |
|
chpsscon3 |
|- ( ( A e. CH /\ ( _|_ ` B ) e. CH ) -> ( A C. ( _|_ ` B ) <-> ( _|_ ` ( _|_ ` B ) ) C. ( _|_ ` A ) ) ) |
3 |
1 2
|
sylan2 |
|- ( ( A e. CH /\ B e. CH ) -> ( A C. ( _|_ ` B ) <-> ( _|_ ` ( _|_ ` B ) ) C. ( _|_ ` A ) ) ) |
4 |
|
ococ |
|- ( B e. CH -> ( _|_ ` ( _|_ ` B ) ) = B ) |
5 |
4
|
adantl |
|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` B ) ) = B ) |
6 |
5
|
psseq1d |
|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` ( _|_ ` B ) ) C. ( _|_ ` A ) <-> B C. ( _|_ ` A ) ) ) |
7 |
3 6
|
bitrd |
|- ( ( A e. CH /\ B e. CH ) -> ( A C. ( _|_ ` B ) <-> B C. ( _|_ ` A ) ) ) |