Is it true that P(x|y,z)=P(x|y) if x and z be independent? Why? I know that if x and z be independent, P(xz) = P(x)P(z). 
I want to know if x and z be independent, can I cancel z from P(x|yz)? Why? 
Thanks. 

Although other responses are useful, but here is another solution with different point of view:
$$
P(x|y,z)=\frac{P(x,y,z)}{P(y,z)}=\frac{P(x,y,z)}{\int_xP(x,y,z)dx}=\frac{P(y|x,z)P(x)P(z)}{\int_xP(y|x,z)P(x)P(z)dx}=\frac{P(y|x,z)P(x)}{\int_xP(y|x,z)P(x)dx}
$$
and the last term is obviously related to $z$, unless $y$ and $z$ be conditionally independent given $x$.
