Step |
Hyp |
Ref |
Expression |
1 |
|
sseqin2 |
|- ( A C_ C <-> ( C i^i A ) = A ) |
2 |
1
|
biimpi |
|- ( A C_ C -> ( C i^i A ) = A ) |
3 |
|
incom |
|- ( C i^i A ) = ( A i^i C ) |
4 |
2 3
|
eqtr3di |
|- ( A C_ C -> A = ( A i^i C ) ) |
5 |
4
|
difeq1d |
|- ( A C_ C -> ( A \ B ) = ( ( A i^i C ) \ B ) ) |
6 |
|
difundi |
|- ( C \ ( ( C \ A ) u. B ) ) = ( ( C \ ( C \ A ) ) i^i ( C \ B ) ) |
7 |
|
dfss4 |
|- ( A C_ C <-> ( C \ ( C \ A ) ) = A ) |
8 |
7
|
biimpi |
|- ( A C_ C -> ( C \ ( C \ A ) ) = A ) |
9 |
8
|
ineq1d |
|- ( A C_ C -> ( ( C \ ( C \ A ) ) i^i ( C \ B ) ) = ( A i^i ( C \ B ) ) ) |
10 |
6 9
|
syl5eq |
|- ( A C_ C -> ( C \ ( ( C \ A ) u. B ) ) = ( A i^i ( C \ B ) ) ) |
11 |
|
indif2 |
|- ( A i^i ( C \ B ) ) = ( ( A i^i C ) \ B ) |
12 |
10 11
|
eqtrdi |
|- ( A C_ C -> ( C \ ( ( C \ A ) u. B ) ) = ( ( A i^i C ) \ B ) ) |
13 |
5 12
|
eqtr4d |
|- ( A C_ C -> ( A \ B ) = ( C \ ( ( C \ A ) u. B ) ) ) |