Is this statement similar to law of total expectations true?

Question

I believe that $\mathbb{E}_x(x \mathbb{E}_{y|x}(y|x))=\mathbb{E}_y(xy)$, and that the proof is as follows for the discrete case.

\begin{align} \mathbb{E}_x(x\mathbb{E}_{y|x}(y|x))&=\mathbb{E}_x(x\sum_y y\:p(y|x))\\ &=\sum_y xy\sum_x p(y|x)p(x)\textrm{ Fubini as in iterated expectations}\\ &=\sum_y xy \:p(y)\\ &=\mathbb{E}_y(xy) \end{align}

This seems right to me, but I wanted to check.