In this paper a new sufficiency theorem for strong minima of an optimal control problem with fixed endpoints and mixed equality state-control constraints is presented. The technique is self-contained in the sense that it does not use the Hamiltonian-Jacobi theory, Riccati equations, fields of extremals, or generalized notions of conjugate points. The absence of the strengthened Legendre-Clebsch condition in the sufficiency statements of the main theorem of this work is a fundamental component.