Independence of premise for intuitionistic Zermelo-Fraenkel set theory

Abstract

We answer the following question by Frittaion, Nemoto, and Rathjen positively: "Is [independence of premise] an admissible rule of $\textsf{CZF}$ or any other familiar constructive/intuitionistic set theory $T$?" To be more precise, we show that whenever $\textsf{IZF}$ (or $\textsf{CZF}$ with full separation) derives a statement of the form $\lnotψ\to \exists y\ φ(y)$ where $y$ does not occur in $ψ$, then $\textsf{IZF}$ (or $\textsf{CZF}$ with full separation) also derives $\exists y\ (\lnotψ\to φ(y))$. Our proof method is a variant of the famous Friedman-Dragalin $A$-translation adapted to the context of set theory in a hereditary manner.