Multivariable ({\phi},{\Gamma})-modules and products of Galois groups

We show that the category of continuous representations of the $d$th direct power of the absolute Galois group of ${\mathbb{Q}}_p$ on finite dimensional ${\mathbb{F}}_p$-vector spaces (resp. finitely generated ${\mathbb{Z}}_p$-modules, resp. finite dimensional ${\mathbb{Q}}_p$-vector spaces) is equivalent to the category of 'etale $(\phi ,\mathrm{\Gamma })$-modules over a $d$-variable Laurent-series ring over ${\mathbb{F}}_p$ (resp. over ${\mathbb{Z}}_p$, resp. over ${\mathbb{Q}}_p$).