All single-stage isobaric flash processes involving multicomponent homogeneous mixtures are shown to have unique two-phase solutions. No simplifications with regard to nonideal phase behavior are made. The main result for the isothermal, isobaric (TP) case is established with the aid of a different characterization of material stability, the Gibbs-Duhem equation, and the Cauchy interlace theorem. Results for the other traditional specifications of heat duty and pressure (QP), and total vapor flow rate and pressure (VP) are proved by establishing a one-to-one correspondence between the solution sets for TP and QP and the solution sets for TP and VP flash problems, respectively. Construction of these one-to-one mappings results in some interesting analysis associated with the appropriate plane curves in the TQ and TV planes. Some new properties of the underlying matrices involved in the phase equilibrium of homogeneous mixtures are presented.