In this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation. We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method's finite element space Mh satisfies two conditions, then the two approximation methods are equivalent. These two conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. For any such mixed spaces defined on a geometrically regular partition of the domain, we can then easily construct appropriate conforming spaces Mh. We also present for several mixed methods alternative nonconforming spaces Mh that also satisfy the two conditions for equivalence.