Step |
Hyp |
Ref |
Expression |
1 |
|
bicom |
|- ( ( th <-> ta ) <-> ( ta <-> th ) ) |
2 |
|
imbi2 |
|- ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) ) |
3 |
2
|
biimpcd |
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) ) |
4 |
1 3
|
mpi |
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) |
5 |
4
|
3expd |
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) |
6 |
|
3impexp |
|- ( ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) |
7 |
6
|
biimpri |
|- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) |
8 |
7 1
|
syl6ibr |
|- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ) |
9 |
5 8
|
impbii |
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) |