We investigate the phase structure of SU(4) gauge theory with the gauge field simultaneously coupled to two flavors of fermions in the fundamental representation and two flavors of fermions in the two-index antisymmetric representation. We find that the theory has only two phases: a low-temperature phase with both species of fermion confined and chirally broken, and a high-temperature phase with both species of fermion deconfined and chirally restored. The single phase transition in the theory appears to be first order, in agreement with theoretical predictions.