Theorem bj-hbxfrbi

Description: Closed form of hbxfrbi . Note: it is less important than nfbiit . The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht ) in order not to require sp (modal T). See bj-hbyfrbi for its version with existential quantifiers. (Contributed by BJ, 6-May-2019)

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

Proof

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