The assumption of independency is about errors of **independently measured** quantities (as there is almost always some function $f$ which connects all the measured quantities), i.e. if you measured P and V independently (for example using different measuring devices) - their errors are independent of all the other measurements, and thus you can use it.

Just to make it clear, the general formula for calculating error of $ f(x_1,x_2,...,x_n)$
(where $x_i$ are independently measured quantities) is $$ \delta f = \sqrt{\sum_{i=1}^{n}(\frac{\partial f}{\partial x_i}\delta x_i)^2}$$

$\delta x_i$ are uncertainties in $x_i$ measurement. All the derivatives are calculated at the point $(x_1,x_2,...,x_n)$ - measured values.