Theorem abbi

Description: Equivalent formulas yield equal class abstractions (closed form). This is the backward implication of abbib , proved from fewer axioms, and hence is independently named. (Contributed by BJ and WL and SN, 20-Aug-2023)

	Ref	Expression
Assertion	abbi	|- ( A. x ( ph <-> ps ) -> { x | ph } = { x | ps } )

Proof

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