Step |
Hyp |
Ref |
Expression |
1 |
|
csbin |
|- [_ A / x ]_ ( D i^i ( `' R " { X } ) ) = ( [_ A / x ]_ D i^i [_ A / x ]_ ( `' R " { X } ) ) |
2 |
|
csbima12 |
|- [_ A / x ]_ ( `' R " { X } ) = ( [_ A / x ]_ `' R " [_ A / x ]_ { X } ) |
3 |
|
csbcnv |
|- `' [_ A / x ]_ R = [_ A / x ]_ `' R |
4 |
3
|
imaeq1i |
|- ( `' [_ A / x ]_ R " [_ A / x ]_ { X } ) = ( [_ A / x ]_ `' R " [_ A / x ]_ { X } ) |
5 |
|
csbsng |
|- ( A e. V -> [_ A / x ]_ { X } = { [_ A / x ]_ X } ) |
6 |
5
|
imaeq2d |
|- ( A e. V -> ( `' [_ A / x ]_ R " [_ A / x ]_ { X } ) = ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) ) |
7 |
4 6
|
eqtr3id |
|- ( A e. V -> ( [_ A / x ]_ `' R " [_ A / x ]_ { X } ) = ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) ) |
8 |
2 7
|
eqtrid |
|- ( A e. V -> [_ A / x ]_ ( `' R " { X } ) = ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) ) |
9 |
8
|
ineq2d |
|- ( A e. V -> ( [_ A / x ]_ D i^i [_ A / x ]_ ( `' R " { X } ) ) = ( [_ A / x ]_ D i^i ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) ) ) |
10 |
1 9
|
eqtrid |
|- ( A e. V -> [_ A / x ]_ ( D i^i ( `' R " { X } ) ) = ( [_ A / x ]_ D i^i ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) ) ) |
11 |
|
df-pred |
|- Pred ( R , D , X ) = ( D i^i ( `' R " { X } ) ) |
12 |
11
|
csbeq2i |
|- [_ A / x ]_ Pred ( R , D , X ) = [_ A / x ]_ ( D i^i ( `' R " { X } ) ) |
13 |
|
df-pred |
|- Pred ( [_ A / x ]_ R , [_ A / x ]_ D , [_ A / x ]_ X ) = ( [_ A / x ]_ D i^i ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) ) |
14 |
10 12 13
|
3eqtr4g |
|- ( A e. V -> [_ A / x ]_ Pred ( R , D , X ) = Pred ( [_ A / x ]_ R , [_ A / x ]_ D , [_ A / x ]_ X ) ) |