Let $X\sim N(\mu,\sigma^2)$. It's straightforward to derive (see, e.g., distribution of the normal cdf and https://mathoverflow.net/questions/225868/variance-of-the-normal-cdf?) $E[\Phi(X)]$ and $Var(\Phi(X))$, where $\Phi(\cdot)$ is the normal CDF.
Is it also straightforward to characterize the full distribution of $\Phi(X)$?
Even more so: for any continuous distribution with CDF $F$:
If $F(x) = y$, $0 < y < 1$, $$F_{F(X)}(y) = \mathbb P(F(X) \le y) = \mathbb P(X \le x) = F(x) = y$$
That is, $F(X)$ has uniform distribution on $[0,1]$.