Step |
Hyp |
Ref |
Expression |
1 |
|
bicom1 |
|- ( ( ph <-> ch ) -> ( ch <-> ph ) ) |
2 |
|
nanbi2 |
|- ( ( ph <-> ch ) -> ( ( ps -/\ ph ) <-> ( ps -/\ ch ) ) ) |
3 |
1 2
|
nanbi12d |
|- ( ( ph <-> ch ) -> ( ( ch -/\ ( ps -/\ ph ) ) <-> ( ph -/\ ( ps -/\ ch ) ) ) ) |
4 |
|
nannan |
|- ( ( ph -/\ ( ps -/\ ch ) ) <-> ( ph -> ( ps /\ ch ) ) ) |
5 |
|
simpr |
|- ( ( ps /\ ch ) -> ch ) |
6 |
5
|
imim2i |
|- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ch ) ) |
7 |
4 6
|
sylbi |
|- ( ( ph -/\ ( ps -/\ ch ) ) -> ( ph -> ch ) ) |
8 |
|
nannan |
|- ( ( ch -/\ ( ps -/\ ph ) ) <-> ( ch -> ( ps /\ ph ) ) ) |
9 |
|
simpr |
|- ( ( ps /\ ph ) -> ph ) |
10 |
9
|
imim2i |
|- ( ( ch -> ( ps /\ ph ) ) -> ( ch -> ph ) ) |
11 |
8 10
|
sylbi |
|- ( ( ch -/\ ( ps -/\ ph ) ) -> ( ch -> ph ) ) |
12 |
7 11
|
impbid21d |
|- ( ( ch -/\ ( ps -/\ ph ) ) -> ( ( ph -/\ ( ps -/\ ch ) ) -> ( ph <-> ch ) ) ) |
13 |
|
nanan |
|- ( ( ph /\ ( ps -/\ ch ) ) <-> -. ( ph -/\ ( ps -/\ ch ) ) ) |
14 |
|
simpl |
|- ( ( ph /\ ( ps -/\ ch ) ) -> ph ) |
15 |
13 14
|
sylbir |
|- ( -. ( ph -/\ ( ps -/\ ch ) ) -> ph ) |
16 |
|
nanan |
|- ( ( ch /\ ( ps -/\ ph ) ) <-> -. ( ch -/\ ( ps -/\ ph ) ) ) |
17 |
|
simpl |
|- ( ( ch /\ ( ps -/\ ph ) ) -> ch ) |
18 |
16 17
|
sylbir |
|- ( -. ( ch -/\ ( ps -/\ ph ) ) -> ch ) |
19 |
|
pm5.1im |
|- ( ph -> ( ch -> ( ph <-> ch ) ) ) |
20 |
15 18 19
|
syl2imc |
|- ( -. ( ch -/\ ( ps -/\ ph ) ) -> ( -. ( ph -/\ ( ps -/\ ch ) ) -> ( ph <-> ch ) ) ) |
21 |
12 20
|
bija |
|- ( ( ( ch -/\ ( ps -/\ ph ) ) <-> ( ph -/\ ( ps -/\ ch ) ) ) -> ( ph <-> ch ) ) |
22 |
3 21
|
impbii |
|- ( ( ph <-> ch ) <-> ( ( ch -/\ ( ps -/\ ph ) ) <-> ( ph -/\ ( ps -/\ ch ) ) ) ) |
23 |
|
nancom |
|- ( ( ps -/\ ph ) <-> ( ph -/\ ps ) ) |
24 |
23
|
nanbi2i |
|- ( ( ch -/\ ( ps -/\ ph ) ) <-> ( ch -/\ ( ph -/\ ps ) ) ) |
25 |
|
nancom |
|- ( ( ch -/\ ( ph -/\ ps ) ) <-> ( ( ph -/\ ps ) -/\ ch ) ) |
26 |
24 25
|
bitri |
|- ( ( ch -/\ ( ps -/\ ph ) ) <-> ( ( ph -/\ ps ) -/\ ch ) ) |
27 |
26
|
bibi1i |
|- ( ( ( ch -/\ ( ps -/\ ph ) ) <-> ( ph -/\ ( ps -/\ ch ) ) ) <-> ( ( ( ph -/\ ps ) -/\ ch ) <-> ( ph -/\ ( ps -/\ ch ) ) ) ) |
28 |
22 27
|
bitri |
|- ( ( ph <-> ch ) <-> ( ( ( ph -/\ ps ) -/\ ch ) <-> ( ph -/\ ( ps -/\ ch ) ) ) ) |