Step |
Hyp |
Ref |
Expression |
1 |
|
ineq1 |
|- ( A = B -> ( A i^i B ) = ( B i^i B ) ) |
2 |
|
inidm |
|- ( B i^i B ) = B |
3 |
1 2
|
eqtrdi |
|- ( A = B -> ( A i^i B ) = B ) |
4 |
3
|
eqeq1d |
|- ( A = B -> ( ( A i^i B ) = (/) <-> B = (/) ) ) |
5 |
|
eqtr |
|- ( ( A = B /\ B = (/) ) -> A = (/) ) |
6 |
|
simpr |
|- ( ( A = B /\ B = (/) ) -> B = (/) ) |
7 |
5 6
|
jca |
|- ( ( A = B /\ B = (/) ) -> ( A = (/) /\ B = (/) ) ) |
8 |
7
|
ex |
|- ( A = B -> ( B = (/) -> ( A = (/) /\ B = (/) ) ) ) |
9 |
4 8
|
sylbid |
|- ( A = B -> ( ( A i^i B ) = (/) -> ( A = (/) /\ B = (/) ) ) ) |
10 |
9
|
com12 |
|- ( ( A i^i B ) = (/) -> ( A = B -> ( A = (/) /\ B = (/) ) ) ) |
11 |
|
eqtr3 |
|- ( ( A = (/) /\ B = (/) ) -> A = B ) |
12 |
10 11
|
impbid1 |
|- ( ( A i^i B ) = (/) -> ( A = B <-> ( A = (/) /\ B = (/) ) ) ) |