Theorem ss2ab1

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 } )

Proof

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 } )