Is the law of iterated expectation correctly applied here?

Question

I am following this video It is established that:

$\mathbf E[\tilde\Theta\hat\Theta\mid X=x] = 0.$

Using the law of iterated expectations,

$\mathbf E\bigl[\mathbf E[X|Y]\bigr] = \mathbf E[X]$

it argues that

$\mathbf E[\tilde\Theta\hat\Theta] = \mathbf E[\tilde\Theta\hat\Theta\mid X=x] = 0.$

However what I deduce from the law of iterated expectation is:

$\mathbf E[\tilde\Theta\hat\Theta] = \mathbf E\bigl[\mathbf E[\tilde\Theta\hat\Theta\mid X=x]\bigr].$

Are they saying

$\mathbf E[\tilde\Theta\hat\Theta\mid X=x] = \mathbf E\bigl[\mathbf E[\tilde\Theta\hat\Theta\mid X=x]\bigr]?$

Is this a violation of the law's definition?

Answer 1

The answer is:

$\mathbf E[\tilde\Theta\hat\Theta\mid X=x] = \mathbf E\bigl[\mathbf E[\tilde\Theta\hat\Theta\mid X=x]\bigr] = \mathbf E[0] = 0.$