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For nonlinear systems, the implementation of a state feedback with observer estimates may lead to severe forms of instability. The standard approach to analyzing observer-based controllers is to treat the observer error as a "measurement error". In this paper we eliminate the conservatism of this approach by explicitly including the observer dynamics in stability analysis. This is achieved with a new detectability concept which, when combined with an additional condition on the Lyapunov function for the underlying full-state feedback design, guarantees stability of certainty-equivalence. An application of our result to strict-feedback systems shows that, under a mild polynomial growth assumption on nonlinearities, stability can be achieved with a certainty-equivalence implementation of full-state backstepping designs.