The error that we introduce when $f$ as $\hat{f}$ is called "reducible" error because it is possible to reduce this error by using more suitable statistical models to estimate $f$. Assuming that we can reduce the reducible error to 0, $\implies \hat{Y}=f(X)$, but we still suffer from the irreducible error, because: $Y=f(X)+\epsilon \implies Y=\hat{Y}+\epsilon$ And we know that $\epsilon$ is independent of $X$. So, regardless of the statistical techniques that we use in estimating $f$, we cannot reduce the irreducible error. $E(Y-\hat{Y})^2=E(f(x)+\epsilon-\hat{f}(X))^2=$