It's obvious that when $X,Y,Z$ are independent, we have $P\{(X|Y)|Z\} = P\{X | (Y \cap Z)\},$ but if we only have $Y$ and $Z$ are independent, does this equation still holds?
Edit:
OK, a bit about how this comes. I saw an attempt to calculate $P\{X|Y\}$ goes like this $ P\{X|Y\} = P\{X|Y \cap Z\}P\{Z\} + P\{X|Y \cap Z^c\}P\{Z^c\}. $ My interpretation of this is that \begin{align*} & P\{X|Y\} = P\{(X|Y) \cap Z\} + P\{(X|Y) \cap Z^c\} \\ & \; = P\{(X|Y) | Z\}P\{Z\} + P\{(X|Y) | Z^c\}P\{Z^c\} \\ & \; = P\{X|Y \cap Z\}P\{Z\} + P\{X|Y \cap Z^c\}P\{Z^c\}. \end{align*}
So I was wondering what is the condition to have
$ P\{ (X|Y) | Z\} = P\{X|Y \cap Z \} $