problem in calculation in 'Deep Learning' book

Question

In Goodfellow's, Bengio's, and Courville's 'Deep Learning', Eq. (5.38), the authors compute: $$\mathbb E \left( \frac1m \sum_{i=1}^m \big( x^{(i)} -\hat{\mu}_m \big)^2 \right) = \frac{m-1}m \sigma^2 $$ Here, $\mathbb E$ is the expected value, $x^{(1)} , \ldots , x^{(m)}$ are i.i.d. Gaussian random variables, $\hat\mu_m = \frac1m \sum_{i=1}^m x^{(i)}$ is the empirical mean.
I tried applying linearity of the expected value together with the definition of $\hat\mu_m$ but that didn't work out. I expect it should be quite easy, given the correct trick.
Any help is highly appreciated.