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I know that the conditional probability $p(x\mid y)$ is defined as $\frac{p(x,y)}{p(y)}$. But what if I have $p(x,y \mid z)$, is that the same as $\frac{p(x,y,z)}{p(z)}$?

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Yes,$$p(x,y|z)=\dfrac{p((x,y),z)}{p(z)}=\dfrac{p(x,y,z)}{p(z)}$$ is identical to the definition you quote by treating $(x,y)$ as a (vectorial) random variable.

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