Although good encryption functions are probabilistic, most symbolic models do not capture this aspect explicitly. A typical solution, recently used to prove the soundness of such models with respect to computational ones, is to explicitly represent the dependency of ciphertexts on random coins as labels. In order to make these label-based models useful, it seems natural to try to extend the underlying decision procedures and the implementation of existing tools. In this paper we put forth a more practical alternative based on the following soundness theorem. We prove that for a large class of security properties (that includes rather standard formulations for secrecy and authenticity properties), security of protocols in the simpler model implies security in the label-based model. Combined with the soundness result of (\textbf{?}) our theorem enables the translation of security results in unlabeled symbolic models to computational security.