It's shown in a research paper that: $$ E\left[ \Phi\left(\frac{\Phi^{-1}(PD)-\sqrt{\rho_{1}}Y}{\sqrt{1-\rho_1}}\right) \times \Phi\left(\frac{\Phi^{-1}(LGD)-\sqrt{\rho_{2}}Z}{\sqrt{1-\rho_2}}\right)\right]= \Phi_2\left( \Phi^{-1}(PD), \Phi^{-1}(LGD); \rho\sqrt{\rho_1\rho_2}\right) $$ where $Y$ and $Z$ are both standard normal distributed random variables and their correlation is $\rho$.
I was trying to derive this equation, but no success. Is there anyone could help? Thank you very much.