Step |
Hyp |
Ref |
Expression |
1 |
|
sbcel12 |
|- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) |
2 |
|
csbconstg |
|- ( A e. _V -> [_ A / x ]_ B = B ) |
3 |
2
|
eleq1d |
|- ( A e. _V -> ( [_ A / x ]_ B e. [_ A / x ]_ C <-> B e. [_ A / x ]_ C ) ) |
4 |
1 3
|
syl5bb |
|- ( A e. _V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) ) |
5 |
|
sbcex |
|- ( [. A / x ]. B e. C -> A e. _V ) |
6 |
5
|
con3i |
|- ( -. A e. _V -> -. [. A / x ]. B e. C ) |
7 |
|
noel |
|- -. B e. (/) |
8 |
|
csbprc |
|- ( -. A e. _V -> [_ A / x ]_ C = (/) ) |
9 |
8
|
eleq2d |
|- ( -. A e. _V -> ( B e. [_ A / x ]_ C <-> B e. (/) ) ) |
10 |
7 9
|
mtbiri |
|- ( -. A e. _V -> -. B e. [_ A / x ]_ C ) |
11 |
6 10
|
2falsed |
|- ( -. A e. _V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) ) |
12 |
4 11
|
pm2.61i |
|- ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) |