Theorem bj-hbyfrbi

Description: Version of bj-hbxfrbi with existential quantifiers. (Contributed by BJ, 23-Aug-2023)

	Ref	Expression
Assertion	bj-hbyfrbi	|- ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( ( E. x ph -> ph ) <-> ( E. x ps -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		exbi	|- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) )
2	1	adantl	|- ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( E. x ph <-> E. x ps ) )
3		simpl	|- ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( ph <-> ps ) )
4	2 3	imbi12d	|- ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( ( E. x ph -> ph ) <-> ( E. x ps -> ps ) ) )