We demonstrate how a specification for the standard evaluation of a simple functional programming language can be systematically extended to a specification for mixed evaluation. Using techniques inspired by natural semantics we specify a standard evaluator by a set of inference rules. The evaluation of programs is then performed by a restricted kind of theorem proving in this logic. We then describe a systematic method for extending the proof system for standard evaluation to a new proof system that provides greater flexibility in treating bound variables in the object-level functional programs. We demonstrate how this extended proof system provides the capabilities of a mixed evaluator and how correctness with respect to standard evaluation can be proved in a simple and direct manner. The current work focuses only on a primitive notion of mixed evaluation for a simple functional programming language, but we believe that our methods will extend to more sophisticated kinds of evaluations and richer languages.