A Functional Identity involving Elliptic Integrals

We show that the following double integral $$\int_{0}^\pi {\rm d}x \int_0^x {\rm d}y \frac{1}{\sqrt{1-\smash[b]{p}\cos x}\sqrt{1+\smash[b]{q\cos y}}}$$remains invariant as one trades the parameters $p$ and $q$ for $p'=\sqrt{1-p^2}$ and $q'=\sqrt{1-q^2}$ respectively. This invariance property is suggested from symmetry considerations in the operating characterstics of a semiconductor Hall-effect device.