Let's assume I have the following random variable
$$p(Y) = \Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho} Y}{\sqrt{1-\rho}}\right) \;\; \text{with} \;\ Y \sim N(0,1)$$ where $\Phi(\cdot)$ denotes the standard normal cumulative distribution function (cdf), $p$ is a probability and $\rho$ is a real number between $0$ and $1$. Hence, we now that $p(Y)$ is a non-linear, strictly decreasing, function of $Y$. Let's assume I want to square this random variable, for example during the computation of the variance, then we should write
$$p(Y)^2 = \left(\Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho} Y}{\sqrt{1-\rho}}\right)\right)^2 = \left(\Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho} Y}{\sqrt{1-\rho}}\right) \times \Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho} Y}{\sqrt{1-\rho}}\right)\right) = \Phi^2\left(\frac{\Phi^{-1}(p)-\sqrt{\rho} Y}{\sqrt{1-\rho}}\right)$$
Does the last equality holds? I have encountered this notation in some textbooks, but I am not really sure about its meaning and possible use in this setup. Can we talk about the squared normal cdf?
I feel the latter can become confusing, especially, when we note as $\Phi_2(\cdot)$ the standard normal bivariate.
Thank you in advance for your help on this!