Question on tower property

Question

in a book of mine it says that for sub $\sigma$-algebras $\mathbb D \subset \mathbb E$ $$ \mathrm E [X \mid \mathbb D] = \mathrm E [ \mathrm E [ X \mid \mathbb E ] \mid \mathbb D ] = \mathrm E [ \mathrm E [ X \mid \mathbb D] \mid \mathbb E ] $$ The first equality I recognize as the tower property, but the later I cannot make sense of. If $\mathbb E \not = \mathbb D$, then I can find $E \in \mathbb E$ subject to $E \not \in \mathbb D$ where it should hold that $$ \int_E \mathrm E [ \mathrm E [ X \mid \mathbb D] \mid \mathbb E ] dP = \int_E E[X \mid \mathbb D] dP $$ but $E[X \mid \mathbb D]$ is only $\mathbb D$-measurable. Can anyone help me out?

Answer 1

Recall that if $X$ is $\mathbb F-$measurable, then $$\mathbb E[X\mid\mathbb F]=X.$$

Since $\mathbb E[X\mid \mathbb D]$ is $\mathbb E-$measurable,

$$\mathbb E[\mathbb E[X\mid \mathbb D]\mid \mathbb E]=\mathbb E[X\mid \mathbb D].$$