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Reliability analysis of phased-mission systems (PMS) must consider the statistical dependences of element states across different phases as well as changes in system configuration, success criteria, and component behavior. This paper proposes a recursive method for the exact reliability evaluation of PMS consisting of nonidentical independent nonrepairable multistate elements. The method is based on conditional probabilities and the branch-and-bound principle. It is invariant to changes in system structure, demand, and the elements' state transition rates among phases. The main advantage of this method is that it does not require the composition of decision diagrams and can be fully automated. Both analytical and numerical examples are presented to illustrate the application and advantages of the proposed method. The computational performance of the proposed algorithm is illustrated through comprehensive experimentation on the CPU running time of the algorithm.