If $P(X|W,Z)=P(X|W), P(Y|W,Z)=P(Y|W)$, then $P((X,Y)|W,Z)=P((X,Y)|W)$?

Question

I am lost with the following question. Assume that $W$, $X$, $Y$ and $Z$ are random variables defined on some probability space such that \begin{align*} P(X=x|W=w,Z=z)&=P(X=x|W=w)\\ P(Y=y|W=w,Z=z)&=P(Y=y|W=w). \end{align*} Is it true that $P((X,Y)=(x,y)|W=w,Z=z)=P((X,Y)=(x,y)|W=w)$? If it is true, how can I prove it? If it is not true, are there conditions for this to be true?