Bounded Φ-variation functions are development and generalization of bounded variation functions in the usual sense. The concept of Henstock-Kurzweil integral is an effective tool in dealing with highly infinite oscillation functions. In this paper, the concept of locally right uniquenees of a discontinuous system is defined for generalized integrals at the sense of Henstock-Kurzweil by using Φ-function theory. The bounded variation solution is generalized to bounded Φ-variation solution, and the Osgood-type uniqueness theorem for this solution of discontinuous system is established.This result is essential generalization of uniqueness for bounded variation solutions of the system, and the certain foundation is laid in the research of highly infinite oscillation functions.