Description: Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005) (Revised by Mario Carneiro, 1-Sep-2015)
Ref | Expression | ||
---|---|---|---|
Hypothesis | csbeq2dv.1 | |- ( ph -> B = C ) |
|
Assertion | csbeq2dv | |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq2dv.1 | |- ( ph -> B = C ) |
|
2 | 1 | eleq2d | |- ( ph -> ( y e. B <-> y e. C ) ) |
3 | 2 | sbcbidv | |- ( ph -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) ) |
4 | 3 | abbidv | |- ( ph -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } ) |
5 | df-csb | |- [_ A / x ]_ B = { y | [. A / x ]. y e. B } |
|
6 | df-csb | |- [_ A / x ]_ C = { y | [. A / x ]. y e. C } |
|
7 | 4 5 6 | 3eqtr4g | |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C ) |