In these lecture notes, a conditional PDF (probability density function) is defined as follows (Def. 4.2.3): $$f(y \mid x) = \frac{f(x, y)}{f_{X}(x)}, \qquad (*)$$ where $(X, Y)$ shall be a continuous bivariate random vector.
$f(y \mid x)$ is a PDF, since $$\int f(y\mid x)dy = \int\frac{f(x, y)}{f_{X}(x)}dy = \frac{f_{X}(x)}{f_{X}(x)} = 1.$$
My question would now be: can this definition be expanded to the case that multiple random variables for the conditioning are given, i.e. could $(*)$ be generalized as follows: $$f(y\mid x_{1}, \dots, x_{n}) = \frac{f(x_1, \dots, x_{n}, y)}{f(x_1, \dots, x_{n})}?\qquad (**)$$ What I'm having in mind is that the $x_i$ come from different random variables $X_i$, where the random variables $X_i$ can follow different PDF's, e.g. $X_1 \sim N(\mu, \sigma)$, $X_2 \sim U(a, b)$ (where the first denotes a normal distribution with mean $\mu$ and variance $\sigma^2$, the latter a uniform distribution with bounds $a$ and $b$).
Personally, I think yes, since when integrating over $y$, we should still get an integral of $1$, but since I haven't seen the definition $(**)$ anywhere, I thought I better ask.
Also, if a reference to the above definition $(**)$ - if existent - could be provided, I'd appreciate it!