Find $Cor(\Phi(X),\Phi(Y)),$ where $X,Y$ are standard normal with $\operatorname{Cor}(X,Y)=\rho$, $ \Phi(x)=\int_{-\infty}^x\phi(x)dx$ and $\phi(x)$ is the standard normal density.
Using the substitution $x=\rho y+\sqrt{1-\rho^2}t$, I was able to simplify it a bit and now need to integrate the following $$E[\Phi(X)\Phi(Y)]=\frac 1{2\pi}\int_{\mathbb R}\int_{\mathbb R}\Phi(\rho y+\sqrt{1-\rho^2}t)\Phi(y)\phi(t)\phi(y)\,dt\,dy$$
Any hints how to proceed? I know $\Phi(X)$ follows uniform distribution
PS. The final answer is $\frac 6\pi\arcsin\frac\rho2$