How do I find probability density function $f_{X,Y}$ of this multivariable function $F_{X,Y}$ where $\Phi$ is normal distribution function, $\alpha \in [0,1]$ and $\mu, \sigma \in \mathbb{R}$ and $$ F_{X,Y} = \alpha \Phi\left(\frac{x-\mu_1}{\sigma_1}\right)\Phi(y) + (1-\alpha) \Phi\left(\frac{y-\mu_2}{\sigma_2}\right)\Phi(x) $$
$f_{X,Y}$ has to be expressed through $\sigma_i, \mu_i$ and $\alpha$. We don't know if these functions are independent in fact we actually later have to find the $\mu_i$ and $\sigma_i$ where they are independent. I'm guessing I have to use the formula $$ f_{X,Y} = \frac{\partial^2 F_{X,Y}}{\partial x \partial y}, $$ but this gets quite ugly in my opinion. I have a feeling that the answer should be simpler.
Hint: $$ f_{X,Y}(x,y) = \frac{1}{\sigma_1}\cdot\alpha \phi\left(\frac{x-\mu_1}{\sigma_1}\right)\phi(y) + \frac{1}{\sigma_2}\cdot(1-\alpha) \phi\left(\frac{y-\mu_2}{\sigma_2}\right)\phi(x) $$ where $\phi$ is the density function of the standard normal distribution $\mathcal{N}(0,1)$