Proving Tychonoff's theorem with the Compactness theorem of logic

Question

It seems to be known that Tychonoff's Theorem for Hausdorff spaces and the Compactness theorem of first order logic are both equivalent over ZF to the ultrafilter lemma. Does anyone know a slick proof for the implication "Compactness Theorem $\rightarrow$ Tychonoff for Hausdorff spaces" (without using the ultrafilter lemma as an intermediate step)?