Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse.

It is evident that $$ \mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p, $$ where $X\sim N(0,1)$.

Is there possible to get any simplified expression of $$ \mathbb{P}(X<\Phi^{-1}(p),\,Y<\Phi^{-1}(p))= \int_{-\infty}^{\Phi^{-1}(p)}\varphi(y)\Phi\left(\frac{-\varrho y+\Phi^{-1}(p)}{\sqrt{1-\varrho^2}}\right)\,dy, $$ where $X,\,Y\sim N(0,1)$, $\textrm{cov}(X,Y)=\varrho$ and $\varphi(y)=\frac{1}{\sqrt{2\pi}}e^{-y^2/2}$?