Theorem ss2abdv

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011) (Revised by Steven Nguyen, 28-Jun-2024)

	Ref	Expression
Hypothesis	ss2abdv.1	|- ( ph -> ( ps -> ch ) )
Assertion	ss2abdv	|- ( ph -> { x | ps } C_ { x | ch } )

Proof

Step	Hyp	Ref	Expression
1		ss2abdv.1	|- ( ph -> ( ps -> ch ) )
2	1	sbimdv	|- ( ph -> ( [ y / x ] ps -> [ y / x ] ch ) )
3		df-clab	|- ( y e. { x | ps } <-> [ y / x ] ps )
4		df-clab	|- ( y e. { x | ch } <-> [ y / x ] ch )
5	2 3 4	3imtr4g	|- ( ph -> ( y e. { x | ps } -> y e. { x | ch } ) )
6	5	ssrdv	|- ( ph -> { x | ps } C_ { x | ch } )