Step |
Hyp |
Ref |
Expression |
1 |
|
idn1 |
|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ). |
2 |
|
bicom |
|- ( ( th <-> ta ) <-> ( ta <-> th ) ) |
3 |
|
imbi2 |
|- ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) ) |
4 |
3
|
biimpcd |
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) ) |
5 |
1 2 4
|
e10 |
|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ). |
6 |
|
3impexp |
|- ( ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) |
7 |
6
|
biimpi |
|- ( ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) |
8 |
5 7
|
e1a |
|- (. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ). |
9 |
8
|
in1 |
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) |
10 |
|
idn1 |
|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ). |
11 |
6
|
biimpri |
|- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) |
12 |
10 11
|
e1a |
|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ). |
13 |
3
|
biimprcd |
|- ( ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) -> ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ) ) |
14 |
12 2 13
|
e10 |
|- (. ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ->. ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ). |
15 |
14
|
in1 |
|- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ) |
16 |
|
impbi |
|- ( ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) -> ( ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) ) -> ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) ) ) |
17 |
9 15 16
|
e00 |
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) ) |