Let random variables $X,Y$ independent fair random variables that take the values 0 and 1 with equal probability and $Z=X+Y$. So, $I(X;Y)=0$ and I am trying to find their conditional mutual information:
\begin{eqnarray*} I(X;Y|Z)&=&H(X|Z)-H(X|Y,Z)\\ &=&H(X|Z)\\ &=&P(Z=1)H(X|Z=1)\\ &=&\frac{1}{2}bit \end{eqnarray*}
I cannot understand how we get to the second line i.e. why $H(X|Y,Z)=0$ and why we don't use joint PMF in the third line.