convolve probit function with gaussian

Question

I want to prove the following, however, not sure where to start.

$\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$

Where $\Phi(\cdot)$ is the probit function, defined as $\Phi(a)=\int_{-\infty}^a\mathcal{N}(x|0,1)dx$

Answer 1

Found answer on https://stats.stackexchange.com/questions/61080/how-can-i-calculate-int-infty-infty-phi-left-fracw-ab-right-phiw . The trick is to consider 2 Gaussian random variables $X$ and $Y$ and consider $X<Y$ is equivalent to $X-Y<0$