Prove that $\{(x,y): W_x\text{ and }W_y\text{ are recursively separable}\}$ is $\Sigma_3$-complete
This is a question from Soare's Recursively Enumerable Sets and Degrees. I have little idea how to construct the reduction function. I have done much simpler completeness proofs of $\Sigma_1$ level but don't really get the intuition behind completeness at higher levels of the hierarchy.