Detailed explanation needed for basic query regarding expectation

Question

I need to find the expectation of following random variable $$g=[\log_2(\frac{1+x}{1+y})]^+$$ where $[x]^+=max(x,0)$ and both $x,y$ variables depend on variable $z$. I know the conditional pdf's and cdf's of $x$ and $y$ conditioned on $z$. I want to know how can we prove the following $$E[g]=\frac{1}{\ln(2)}\int_0^\infty \int_0^\infty \frac{F_{y|z}(x_2)}{1+x_2}(1-F_{x|z}(x_2))f_{z}(x_1)dx_2dx_1$$ where $F's$ are the cdf and $f's$ are pdf. Any help in this regard will be much appreciated. Thank you in advance.