# invariant neighbourhood under a continuous group action

I found this statement and I am really struggling trying to come up with a proof of it. The situation is the following:

Let $$G$$ be a compact topological group acting continuously on a compact Hausdorff space $$X$$. Then for any $$x \in X$$ there is a $$G$$-invariant neighbourhood $$U$$ of $$X$$ such that for any $$y \in U$$, $$gG_yg^{-1} \subseteq G_x$$ for some $$g \in G$$. (Here $$G_x$$ denotes the stabilizer subgroup of $$X$$)

I think that the slice theorem might be useful here (There is a $$G$$-invariant neighbourhood of $$X$$ of the form $$(G \times A)/G_x$$ for some $$G_x$$-invariant subset $$A$$ of $$X$$) but I do not see how to proceed using this.