Under the same regularity conditions as the posterior concentration theorem, with $\hat\theta_n$ denoting the MLE, the posterior distribution of $\sqrt{n}(\theta - \hat\theta_n)$ given $X_1, \ldots, X_n$ converges in total variation to $N(0, I(\theta_0)^{-1})$ in $P_{\theta_0}$-probability. That is, letting $\Pi_n$ denote the posterior and $G_n$ the $N(\hat\theta_n, I(\theta_0)^{-1}/n)$ distribution, \begin{align*} \sup_{A \in \mathcal{B}(\mathbb{R}^p)} \left|\Pi_n(A \mid X_1, \ldots, X_n) - G_n(A)\right| \xrightarrow{\mathbb{P}} 0. \end{align*}