Theorem wl-clabtv

Description: Using class abstraction in a context, requiring x and ph disjoint, but based on fewer axioms than wl-clabt . (Contributed by Wolf Lammen, 29-May-2023)

	Ref	Expression
Assertion	wl-clabtv	|- ( ph -> { x | ps } = { x | ( ph -> ps ) } )

Proof

Step	Hyp	Ref	Expression
1		biimt	|- ( ph -> ( ps <-> ( ph -> ps ) ) )
2	1	sbbidv	|- ( ph -> ( [ y / x ] ps <-> [ y / x ] ( ph -> ps ) ) )
3		df-clab	|- ( y e. { x | ps } <-> [ y / x ] ps )
4		df-clab	|- ( y e. { x | ( ph -> ps ) } <-> [ y / x ] ( ph -> ps ) )
5	2 3 4	3bitr4g	|- ( ph -> ( y e. { x | ps } <-> y e. { x | ( ph -> ps ) } ) )
6	5	eqrdv	|- ( ph -> { x | ps } = { x | ( ph -> ps ) } )