In the context of strategic games, we provide an axiomatic proof of the statement Common knowledge of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies. Rationality here means playing only strategies one believes to be best responses. This involves looking at two formal languages. One is first-order, and is used to formalise optimality conditions, like avoiding strictly dominated strategies, or playing a best response. The other is a modal fixpoint language with expressions for optimality, rationality and belief. Fixpoints are used to form expressions for common belief and for iterated elimination of non-optimal strategies.