Admissibility of independence-of-premise rule for existential quantification over finite types 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^σ φ(y), infer T ⊢ ∃y^σ (¬ψ → φ(y)), for an arbitrary finite type σ and any formula ψ in which y is not free.

Background

The paper proves closure under several independence-of-premise rules for intuitionistic set theories via generic realizability with truth, using reflexive partial combinatory algebras that contain all finite types. Specifically, Theorem Main (ii) establishes a weaker rule where the extracted witness is guaranteed to be of the specified finite type only when the negated premise holds, rather than producing a term of that type uniformly.

The authors explicitly remark that their approach has shortcomings as seen in Theorem Main (ii) and raise the question of whether the genuine independence-of-premise rule with existential quantification over a fixed finite type can be derived as an admissible rule in CZF or similar systems. Establishing this would align the set-theoretic setting with the classical admissibility results known in higher-type arithmetic under modified realizability.