Say there are two continuous random random variables $X$ and $Y$ and I want to find $P(X>Y)$
I know that $P(X>Y)=\int_{o}^{\infty}\int_{0}^{x}f_{xy}(x,y)dydx$
but could I also say that $P(X>Y)=\int_{y}^{\infty}\int_{0}^{\infty}f_{xy}(x,y)dydx$
Reason being that since $X>Y$, $y$ can be any value but when we integrate with respect to $x$, we want to integrate over values greater than $y$.
The second equation should be written as $P\{X>Y\}=\int_0^{\infty} \int _y ^{\infty} f_{xy} (x,y) \, dx \, dy$. What you have written makes RHS dependent on $y$, but it is supposed to be a constant.