Description: Class abstractions in a subclass relationship, closed form. One direction of ss2ab using fewer axioms. (Contributed by SN, 22-Dec-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | ss2ab1 | |- ( A. x ( ph -> ps ) -> { x | ph } C_ { x | ps } ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbim | |- ( A. x ( ph -> ps ) -> ( [ t / x ] ph -> [ t / x ] ps ) ) |
|
2 | df-clab | |- ( t e. { x | ph } <-> [ t / x ] ph ) |
|
3 | df-clab | |- ( t e. { x | ps } <-> [ t / x ] ps ) |
|
4 | 1 2 3 | 3imtr4g | |- ( A. x ( ph -> ps ) -> ( t e. { x | ph } -> t e. { x | ps } ) ) |
5 | 4 | ssrdv | |- ( A. x ( ph -> ps ) -> { x | ph } C_ { x | ps } ) |