Theorem ss2abdvALT

Description: Alternate proof of ss2abdv . Shorter, but requiring ax-8 . (Contributed by Steven Nguyen, 28-Jun-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ss2abdvALT.1	|- ( ph -> ( ps -> ch ) )
Assertion	ss2abdvALT	|- ( ph -> { x | ps } C_ { x | ch } )

Proof

Step	Hyp	Ref	Expression
1		ss2abdvALT.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 } )