On the independence of premiss axiom and rule

In this paper, we deal with a relationship among the law of excluded middle, the double negation elimination and the independence of premiss rule ( $$\mathrm {IPR}$$ IPR ) for (many-sorted) intuitionistic predicate logic. After giving a general machinery, we give, as corollaries, several examples of extensions of $$\mathbf {HA}$$ HA and $$\mathbf {HA}^\omega $$ HA ω which are closed under $$\mathrm {IPR}$$ IPR but do not derive the independence of premiss axiom.