Step |
Hyp |
Ref |
Expression |
1 |
|
elpwi |
|- ( A e. ~P B -> A C_ B ) |
2 |
|
ssidd |
|- ( A C_ B -> A C_ A ) |
3 |
|
id |
|- ( A C_ B -> A C_ B ) |
4 |
2 3
|
ssind |
|- ( A C_ B -> A C_ ( A i^i B ) ) |
5 |
|
ssexg |
|- ( ( A C_ ( A i^i B ) /\ ( A i^i B ) e. V ) -> A e. _V ) |
6 |
4 5
|
sylan |
|- ( ( A C_ B /\ ( A i^i B ) e. V ) -> A e. _V ) |
7 |
|
elpwg |
|- ( A e. _V -> ( A e. ~P B <-> A C_ B ) ) |
8 |
7
|
biimparc |
|- ( ( A C_ B /\ A e. _V ) -> A e. ~P B ) |
9 |
6 8
|
syldan |
|- ( ( A C_ B /\ ( A i^i B ) e. V ) -> A e. ~P B ) |
10 |
9
|
expcom |
|- ( ( A i^i B ) e. V -> ( A C_ B -> A e. ~P B ) ) |
11 |
1 10
|
impbid2 |
|- ( ( A i^i B ) e. V -> ( A e. ~P B <-> A C_ B ) ) |