  Independence of premise principles play an important role in characterizing
the modified realizability and the Dialectica interpretations. In this paper we
show that a great many intuitionistic set theories are closed under the
corresponding independence of premise rule for finite types over $\mathbb{N}$.
It is also shown that the existence property (or existential definability
property) holds for statements of the form $\neg A\to \exists x^{\sigma}
F(x^{\sigma})$, where the variable $x^{\sigma}$ ranges over a finite type
$\sigma$. This applies in particular to Constructive Zermelo-Fraenkel Set
Theory (CZF) and Intuitionistic Zermelo-Fraenkel Set Theory (IZF), two systems
known not to have the general existence property.
  On the technical side, the paper uses the method of realizability with truth
from [21] and [8] with the underlying partial combinatory algebra (pca) chosen
among the total ones. A particular instance of the latter is provided by the
substructure of the graph model formed by the semi computable subsets of
$\mathbb{N}$, which has the advantage that it forms a set pca even in
proof-theoretically weak set theories such as CZF.
