Three random variables, two are independent

Question

Suppose we have 3 random variables $X,Y,Z$ such that $Y\perp Z$ and let $W = X+Y$. How can we infer from this that $$\int f_{WXZ}(x+y,x,z)\mathrm{d}x = \int f_{WX}(x+y,x)f_{Z}(z)\mathrm{d}x$$ Any good reference where I could learn about independence relevant to this question is also welcome.

Answer 1

After integrating out $x$, the left-hand side is the joint density for $Y$ and $Z$ at $(y,z)$, and the right-hand side is the density for $Y$ at $y$ multiplied by the density for $Z$ at $z$. These are the same because $Y$ and $Z$ were assumed to be independent.