Summary: It is shown that the Craig interpolation property and the Beth property are preserved under passage from a superintuitionistic predicate logic to its extension via standard axioms for equality, and under adding formulas of pure equality as new axioms. We find an infinite and independent set of formulas which, though not equivalent to formulas of pure equality, may likewise be added as new axiom schemes without loss of the interpolation, or Beth, property. The formulas are used to construct a continuum of logics with equality, which are intermediate between the intuitionistic and classical ones, having the interpolation property. Moreover, an equality-free fragment of the logics constructed is an intuitionistic predicate logic, and formulas of pure equality satisfy all axioms of the classical predicate logic.