Suppose there is single sample $X_1\dots X_n$ of binary variable $X\sim Bern(p_1)$ and the following hypothesis is being tested:

$$H_0: p_1 = p $$ $$H_1: p_1 \not= p $$

Let us use z-statistic without continuity correction:

$$z = \frac{\hat p -p}{\sqrt{\hat p \hat q / n}}, $$

where $\hat p = \frac 1n \sum X_i,$ $\hat q = 1-\hat p$.

Fleiss in "Statistical Methods for Rates and Proportions" on page 26 states that $z$ is normal just "thanks to the CLT". But this is not obvious to me, because the denominator is also a random variable and we can't use the CLT directly.

Please tell me where can I read about this fact?