How is conditional density function with two given conditions ($f_{X\mid Y,Z}(x\mid y,z)$) defined?

Question

Let $X$, $Y$ and $Z$ be random variables.

Given this conditional density function with two conditions; $Y=y$ and $Z=z$:

$$ f_{X\mid Y,Z}(x \mid y, z) = f_{X\mid Y,Z}(x \mid Y=y, Z=z) $$

I have a hard time understanding this case of two given conditions. I'm trying to replace this expression with one or more other density functions which I'm more familiar.

How do I write $f_{X\mid Y,Z}(x\mid y,z)$ in terms of $f_{X}(x)$, $f_{Y}(y)$, $f_{Z}(z)$, $f_{XY}(x,y)$, $f_{XZ}(x,z)$, $f_{XY\mid Z}(x,y\mid z)$, $f_{XZ\mid Y}(x,z\mid y)$ and $f_{XYZ}(x,y,z)$?

Please write as much identities as possible.

Answer 1

$f_{X\vert Y,Z}(x \vert y,z)= \frac {f_{XYZ}(x,y,z)}{f_{YZ}(y,z)}$ (and $f_{YZ}(y,z) = \int f_{XYZ}(x,y,z) dx$ naturally).