In this paper, we have proved a fundamental property of the characteristic function for the random variable $(\partial/\partial\theta) \log f(x \mid \theta)$. Based on this result, we have proved under regularity conditions different from Bahadur's that certain classes of consistent estimators $\{\theta_n^\ast\}$ are asymptotically efficient in Bahadur's sense $\lim_{\varepsilon \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n\varepsilon^2} \log P\theta\{|\theta_n^\ast - \theta| \geqq \varepsilon\} = -\frac{I(\theta)}{2}.$ Our proof also gives a simple and direct method to verify Bahadur's [2] result.