This paper gives some sufficient conditions for admissible rules to be derivable in intuitionistic propositional calculus. For example, if the premises are Harrop formulas, the rule is admissible only if it is derivable.In deriving the results, a particular class of substitutes is introduced, which are also useful when dealing with other questions of admissibility.

This paper contains a proof theoretic treatment of some aspects of unification in intermediate logics. It is shown that many existing results can be extended to fragments that at least contain implication and conjunction. For such fragments the connection between valuations and most general unifiers is clarified, and it is shown how from the closure of a formula under the Visser rules a proof of the formula under a projective unifier can be obtained. This implies that in the logics considered, for the n-unification type to be finitary it suffices that the m-th Visser rule is admissible for a sufficiently large m. At the end of the paper it is shown how these results imply several well-known results from the literature.

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