Step |
Hyp |
Ref |
Expression |
1 |
|
disjif.1 |
|- F/_ x C |
2 |
|
disjif.2 |
|- ( x = Y -> B = C ) |
3 |
|
df-ne |
|- ( x =/= Y <-> -. x = Y ) |
4 |
|
disjors |
|- ( Disj_ x e. A B <-> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) ) |
5 |
|
equequ1 |
|- ( y = x -> ( y = z <-> x = z ) ) |
6 |
|
csbeq1 |
|- ( y = x -> [_ y / x ]_ B = [_ x / x ]_ B ) |
7 |
|
csbid |
|- [_ x / x ]_ B = B |
8 |
6 7
|
eqtrdi |
|- ( y = x -> [_ y / x ]_ B = B ) |
9 |
8
|
ineq1d |
|- ( y = x -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = ( B i^i [_ z / x ]_ B ) ) |
10 |
9
|
eqeq1d |
|- ( y = x -> ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) <-> ( B i^i [_ z / x ]_ B ) = (/) ) ) |
11 |
5 10
|
orbi12d |
|- ( y = x -> ( ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) <-> ( x = z \/ ( B i^i [_ z / x ]_ B ) = (/) ) ) ) |
12 |
|
eqeq2 |
|- ( z = Y -> ( x = z <-> x = Y ) ) |
13 |
|
nfcv |
|- F/_ x Y |
14 |
13 1 2
|
csbhypf |
|- ( z = Y -> [_ z / x ]_ B = C ) |
15 |
14
|
ineq2d |
|- ( z = Y -> ( B i^i [_ z / x ]_ B ) = ( B i^i C ) ) |
16 |
15
|
eqeq1d |
|- ( z = Y -> ( ( B i^i [_ z / x ]_ B ) = (/) <-> ( B i^i C ) = (/) ) ) |
17 |
12 16
|
orbi12d |
|- ( z = Y -> ( ( x = z \/ ( B i^i [_ z / x ]_ B ) = (/) ) <-> ( x = Y \/ ( B i^i C ) = (/) ) ) ) |
18 |
11 17
|
rspc2v |
|- ( ( x e. A /\ Y e. A ) -> ( A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) -> ( x = Y \/ ( B i^i C ) = (/) ) ) ) |
19 |
4 18
|
syl5bi |
|- ( ( x e. A /\ Y e. A ) -> ( Disj_ x e. A B -> ( x = Y \/ ( B i^i C ) = (/) ) ) ) |
20 |
19
|
impcom |
|- ( ( Disj_ x e. A B /\ ( x e. A /\ Y e. A ) ) -> ( x = Y \/ ( B i^i C ) = (/) ) ) |
21 |
20
|
ord |
|- ( ( Disj_ x e. A B /\ ( x e. A /\ Y e. A ) ) -> ( -. x = Y -> ( B i^i C ) = (/) ) ) |
22 |
3 21
|
syl5bi |
|- ( ( Disj_ x e. A B /\ ( x e. A /\ Y e. A ) ) -> ( x =/= Y -> ( B i^i C ) = (/) ) ) |
23 |
22
|
3impia |
|- ( ( Disj_ x e. A B /\ ( x e. A /\ Y e. A ) /\ x =/= Y ) -> ( B i^i C ) = (/) ) |