Conditional Variance of a Random Variable?

Question

Say I have the following random variable:

Where $\varphi_{i}$ and $\rho_{j}$ are themselves independent random variables (note: given particular values for $\varphi_{i}$ and $\rho_{j}$, $Q_{ij}$ is a Bernoulli random variable with parameter $\varphi_{i}\rho_{j}$). Find,

Here is what I have (I used the law of total variance):

Is my reasoning correct? For some reason I am not confident in it?