equalities between conditional expectation

Question

How can I make the following equation correct by changing the order of $X,Y$ and $E$ : $$E[X]=E[X|E[Y]]$$ and how can I prove it ?

Answer 1

$\mathbb E[Y]$ is a constant, hence \begin{align*} \mathbb E[X|E[Y]] &= \mathbb E[X|\sigma(E[Y])]\\ &=\mathbb E[X|\{\emptyset,\Omega \}]\\ &=\mathbb E[X] \end{align*}

The last line is because any $\{\emptyset,\Omega \}$ measurable random variable is a constant so that $\mathbb E[X|\{\emptyset,\Omega \}]$ is the constant $a$ that satisfies $\mathbb E[Xb]=\mathbb E[ab]$ for all constant $b$, now whenever $b\neq 0$, then we directly get $\mathbb E[X]=a$ by linearity of expectation.