We want to show the following relationship:

If we expand the first expression on the right-hand side of Equation 1:

If we note that $\mathbb E[X]$ is a constant, then $\mathbb E[\mathbb E[X]]$ is also a constant, namely $\mathbb E[X]$. Taking this into account Equation 2 simplifies to:

Plugging Equation 3 back into the right-hand side of Equation 1 gives

thus giving the desired result.