IF$\displaystyle X \;and\; Y$ are independent continuous random variables with $\displaystyle identical \;but\; unknown$ distributions.how can I calculate $\displaystyle P(X>Y)$.

I tried using this concept from the independence and conditional distribution:

$\displaystyle P(X>Y)= E[P(X>Y)|Y] = \displaystyle \int P(X>Y) \; f(y)\;dy = \displaystyle \int (1-F_{X}(y))\; f(y)\;dy$

is this correct?