Proof of propensity score theorem

Question

I'm reading mostly harmless econometrics, and I'm struggling with the proof of the propensity score theorem, you can find it here on page 60: ftp://nozdr.ru/biblio/kolxo3/G/GL/Angrist%20J.D.,%20Pischke%20J.-S.%20Mostly%20Harmless%20Econometrics%20(PUP,%202008)(ISBN%20069112034X)(O)(290s)GL.pdf

Now, in that proof they claim that: $E(E(d_i|y_{ij},p(X_i),X_i)|y_{ji},p(X_i))=E(E(d_i|y_{ji},X_i)|y_{ji},p(X_i))$

I have tried to figure this out with the properties of conditional expectation on Wikipedia. I thought the tower property might help. It says: $E(E(X|Y,Z)|Y)=E(X|Y)$. Yet this suggests that you cant simply write: $E(E(X|Z,Y)|Y)=E(E(X|Z)|Y)$. Can anybody help me?

Answer 1

It has nothing to do with the tower property. It is simply $E(d_i|y_{ij},p(X_i),X_i)=E(d_i|y_{ji},X_i)$ since given $X$, $p(X)$ is redundant.