The implication of law of iterated expectations

Question

Law of iterated expectation: E[Y]=E[E[Y|X]]

As $E[Y|X]$ is already a constant, would that mean that $E[Y]=E[E[Y|X]] = E[Y|X]$?

So $E[Y]=E[Y|X]$?

Or is there something I missed?

Thanks in advance.

Answer 1

$E[Y|X]$ isn't a constant. It's a 'function' that depends on x: $E[Y|X=x]$. If you take the mean of $E[Y|X]$, the mean is taken over all values of x.