I've don't recall seeing a book that gives the probability law
[itex] P((X|Y)|Z) = P(X | (Y \cap Z) ) [/itex]
but I think its true. How to argue it depends on whether you are approaching probability as measure theory or something simpler.
You can also apply the usual probability laws with a condition lilke "[itex]| Z[/itex]" tagged onto every term. For example,
[itex] P(A \cap B) = P(A|B) P(B) [/itex]
[itex] P(A \cap B | Z) = P( (A|B)|Z) P(B | Z) [/itex]
See if you can make progress by applying those ideas.