Can we extend the idea of contour independence for complex contour integrals to several complex variables?

Question

That is, given some function $f:\mathbb{C}^n \to \mathbb{C}$ entire/ sufficiently holomorphic, if we have two domains $D,D'$ in $\mathbb{C}^n$ with the same boundary i.e $\delta D = \delta D' $, will we have $$\int_D f = \int_{D'} f$$ I'm aware there is an analogue for cauchy's theorem on a polydisc, which I would hope we could use in proving this. I've found a proof for the $n=2$ case using Stoke's theorem, but I'm lacking knowledge in the generalised form for stokes and not sure how I could apply it here.