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Conditional independence of random variable $Y$ of $Z$ conditional $X$ is defined as

$$P(Y|X,Z) = P(Y|X)$$

Does this relation hold for any continuous $X$ and $Z$ if $Z = a + bX$, where $a$ and $b$ constants in $\mathbb{R}$? If not, does it hold under specific assumptions?

Edit: A relevant extension of my question is, what about the reverse situation:

$$P(Y|X,Z) = P(Y|Z)?$$

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It will always hold if $Z=f(X)$, i.e. if $Z$ is known given $X$. Apart from that then it depends on $Y$. For example $Y$ independent of $Z$ would work regardless of the relationship between $X$ and $Z$.

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    Yes, this follows from directed graph theory, for example, I can see now. I realize I was more interested in the reverse situation, which is perhaps more interesting also from a graphical models point of view. See edit. – 2017-02-22