It is basic knowledge that we can write $P(a \leq x \leq b) = P(x \leq b) - P(x \leq a)$. But what if we have a multivariate distribution?
Here I know $P(Z \leq Z_x, Z<Z_y) = \Phi_2(Z_x,Z_y,\rho)$ where $\rho$ is the correlation parameter and $\Phi_2$ is the bivariate cumulative normal distribution function. So, how can I write: $P(a \leq Z_x \leq b, c \leq Z_y \leq d)$ with the use of $\Phi_2(Z_x,Z_y,\rho)$?
If $X$ and $Y$ have joint distribution then: $$\mathsf P(a<X\leq b,c<Y\leq d)=F_{X,Y}(b,d)-F_{X,Y}(a,d)-F_{X,Y}(b,c)+F_{X,Y}(a,c)$$