Step |
Hyp |
Ref |
Expression |
1 |
|
excxor |
|- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) |
2 |
1
|
orbi2i |
|- ( ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) <-> ( ( ph /\ ps ) \/ ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) ) |
3 |
|
orc |
|- ( ph -> ( ph \/ ps ) ) |
4 |
|
exmid |
|- ( ps \/ -. ps ) |
5 |
|
pm3.2 |
|- ( ph -> ( ps -> ( ph /\ ps ) ) ) |
6 |
|
biimp |
|- ( ( ph <-> ps ) -> ( ph -> ps ) ) |
7 |
|
iman |
|- ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) |
8 |
6 7
|
sylib |
|- ( ( ph <-> ps ) -> -. ( ph /\ -. ps ) ) |
9 |
8
|
con2i |
|- ( ( ph /\ -. ps ) -> -. ( ph <-> ps ) ) |
10 |
9
|
ex |
|- ( ph -> ( -. ps -> -. ( ph <-> ps ) ) ) |
11 |
|
df-xor |
|- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) |
12 |
11
|
bicomi |
|- ( -. ( ph <-> ps ) <-> ( ph \/_ ps ) ) |
13 |
10 12
|
syl6ib |
|- ( ph -> ( -. ps -> ( ph \/_ ps ) ) ) |
14 |
5 13
|
orim12d |
|- ( ph -> ( ( ps \/ -. ps ) -> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) ) |
15 |
4 14
|
mpi |
|- ( ph -> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) |
16 |
3 15
|
2thd |
|- ( ph -> ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) ) |
17 |
|
bicom |
|- ( ( ph <-> ps ) <-> ( ps <-> ph ) ) |
18 |
|
bibif |
|- ( -. ph -> ( ( ps <-> ph ) <-> -. ps ) ) |
19 |
17 18
|
syl5bb |
|- ( -. ph -> ( ( ph <-> ps ) <-> -. ps ) ) |
20 |
19
|
con2bid |
|- ( -. ph -> ( ps <-> -. ( ph <-> ps ) ) ) |
21 |
20 12
|
syl6bb |
|- ( -. ph -> ( ps <-> ( ph \/_ ps ) ) ) |
22 |
|
biorf |
|- ( -. ph -> ( ps <-> ( ph \/ ps ) ) ) |
23 |
|
simpl |
|- ( ( ph /\ ps ) -> ph ) |
24 |
|
biorf |
|- ( -. ( ph /\ ps ) -> ( ( ph \/_ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) ) |
25 |
23 24
|
nsyl5 |
|- ( -. ph -> ( ( ph \/_ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) ) |
26 |
21 22 25
|
3bitr3d |
|- ( -. ph -> ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) ) |
27 |
16 26
|
pm2.61i |
|- ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) |
28 |
|
3orass |
|- ( ( ( ph /\ ps ) \/ ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) <-> ( ( ph /\ ps ) \/ ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) ) |
29 |
2 27 28
|
3bitr4i |
|- ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) |