Rewrite theories are a general and expressive formalism for specifying concurrent systems in which states are axiomatized by equations and transitions among states are axiomatized by rewrite rules. We present a deductive approach for verifying safety properties of rewrite theories in which all formal temporal reasoning about concurrent transitions is ultimately reduced to purely equational inductive reasoning. Narrowing modulo axioms is extensively used in our inference system to further simplify the equational proof obligations to which all proofs of safety formulas are ultimately reduced. In this way, existing equational reasoning techniques and tools can be directly applied to verify safety properties of concurrent systems. We report on the implementation of this deductive system in the Maude Invariant Analyzer tool, which provides a substantial degree of automation and can automatically discharge many proof obligations without user intervention.