On the elementary theory of the real exponential field

Assuming Schanuel's conjecture, we prove that the complete theory $T_{\exp}$ of the real exponential field is axiomatized by the axioms of definably complete exponential fields satisfying $\exp' = \exp$. This implies the result of Macintyre and Wilkie that, under the same conjecture, $T_{\exp}$ is decidable. Our approach is based on the model completeness of a similar set of axioms for the exponential function restricted to $(-1,1)$, which we prove unconditionally.