Step |
Hyp |
Ref |
Expression |
1 |
|
atsseq |
|- ( ( B e. HAtoms /\ A e. HAtoms ) -> ( B C_ A <-> B = A ) ) |
2 |
|
eqcom |
|- ( B = A <-> A = B ) |
3 |
1 2
|
syl6bb |
|- ( ( B e. HAtoms /\ A e. HAtoms ) -> ( B C_ A <-> A = B ) ) |
4 |
3
|
ancoms |
|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( B C_ A <-> A = B ) ) |
5 |
4
|
necon3bbid |
|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( -. B C_ A <-> A =/= B ) ) |
6 |
|
atelch |
|- ( A e. HAtoms -> A e. CH ) |
7 |
|
atnssm0 |
|- ( ( A e. CH /\ B e. HAtoms ) -> ( -. B C_ A <-> ( A i^i B ) = 0H ) ) |
8 |
6 7
|
sylan |
|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( -. B C_ A <-> ( A i^i B ) = 0H ) ) |
9 |
5 8
|
bitr3d |
|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A =/= B <-> ( A i^i B ) = 0H ) ) |