The logic of single-conclusion (functional) proofs () is introduced. It combines the verification property of proofs with the single valuedness of proof predicate and describes the operations on proofs induced by modus ponens rule and proof checking. It is proved that is decidable, sound and complete with respect to arithmetical proof interpretations based on single-valued proof predicates. The application to arithmetical inference rules specification and -admissibility testing is considered. We show that the provability in gives the complete admissibility test for the rules which can be specified by schemes in -language. The test is supplied with the ground proof extraction algorithm which eliminates the admissible rules from a -proof by utilizing the information from the corresponding -proofs.