In this paper, we show that an intuitionistic logic with second-order function quantification, called hh² here, can serve as a meta-language to directly and naturally specify both sequent calculi and natural deduction inference systems for first-order logic. For the intuitionistic subset of first-order logic, we present a set of hh² formulas which simultaneously specifies both kinds of inference systems and provides a direct and concise account of the correspondence between cut-free sequential proofs and normal natural deductions proofs. The logic of hh² can be implemented using such logic programming techniques as providing operational interpretations to the connectives and implementing unification on lambda-terms. With respect to such an interpreter, our specification provides a description of how to convert a proof in one system to a proof in the other. The operation of converting a sequent proof to a natural deduction proof is functional in the sense that there is always one natural deduction proof corresponding to each sequent proof. Our specification, in fact, provides a direct implementation of the transformation in this direction.