Some new calculi [1,12,7], referred to by the collective name of - calculus, have been recently introduced to provide an explicit treatment of substitutions in the -calculus. They are term rewriting systems, with two sorts: substitution and term. The -terms are exactly the ground -terms of sort term containing no substitutions and the -reduction is decomposed in these calculi, into a starting reduction with a rule called (Beta) followed by a derivation computing explicitly the substitution. These calculi differ by their treatment of substitution. In this paper, we extend the -calculi with a conditional rewriting relation, called c. This relation coincides, on -terms, with the classical-reduction of -calculus. We prove that the confluent -calculus, augmented byc, remains confluent and that the ground confluent version [1], extended by c, is still ground confluent. The result extends readily to Categorical Combinatory Logic. The proof is done by the interpretation method introduced in [9].