Admissibility of independence-of-premise rule without type restrictions in constructive set theories

Determine whether the following rule is admissible in Constructive Zermelo-Fraenkel set theory (CZF) or any other familiar constructive/intuitionistic set theory T: From T ⊢ ¬ψ → ∃y φ(y), infer T ⊢ ∃y (¬ψ → φ(y)), for any formula ψ in which y is not free.

Background

Beyond finite types, the paper shows that certain independence-of-premise rules hold when the existential quantifier is bounded by a finite type, but it does not establish the general untyped version. The general independence-of-premise rule—moving existential quantifiers across an implication with a negated premise—is a core property linked to existence properties and functional interpretations in constructive settings.

The authors explicitly state that the untyped version "remains an open problem" for CZF and similar set theories, motivating further investigation to match known admissibility results in other frameworks or to clarify inherent limitations of set-theoretic realizability with truth.