This paper deals with a complete analysis of a density-dependent model in a chemostat describing the competition of two species for a single nutrient with a mutual-inhibitory relationship. In the presence of species mortality and under general growth functions, we give a quite comprehensive analysis of the existence and local stability of all steady states of the three-dimensional system. The nullcline method permits us to show that if a positive steady state exists, then it is unique and unstable. In this case, the system exhibits a bi-stability where the behavior of the process depends on the initial condition. Our mathematical analysis proves that at most one species can survive which confirms the competitive exclusion principle. Comparing with the previous study of the inter and intraspecific model, we conclude that adding only interspecific competition in the classical chemostat model is not sufficient to show the coexistence of two species even considering mortality in the dynamics of two species.