 Some intermediate predicate logics, extending \(\mathbf{IQC}\) and closed under substitution, are \(\mathbf{IQC}\) \(+\) DNS (Section 4.1), \(\mathbf{IQC}\) \(+\) MP (cf. Section 5.2), \(\mathbf{IQC}\) \(+\) MP \(+\) IP (cf. Section 4.2), and the intuitionistic logic of constant domains \(\mathbf{CD}\) obtained by adding to \(\mathbf{IQC}\) the schema \(\forall x (A \vee B(x)) \rightarrow (A \vee \forall x B(x))\) for all formulas \(A,\) \(B(x)\) with \(x\) not occurring free in \(A.\) Mints, Olkhovikov and Urquhart [2013] showed that \(\mathbf{CD}\) does not have the interpolation property, refuting earlier published proofs by other authors.