(This is based on [my earlier question](https://mathoverflow.net/questions/73815/does-zf-prove-that-topological-groups-are-completely-regular), but I think this one would be easier to answer.)
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Let $\langle X,\mathbf{\delta} \hspace{.01 in} \rangle$ be a [separated proximity space](http://en.wikipedia.org/wiki/Proximity_space), and let $\cal{T}\hspace{.04 in}$ be the induced topology on $X$.
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Then, ZF proves that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is regular Hausdorff, and
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ZF + (Dependent Choice) $\;$proves that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is completely regular.

Does ZF prove that $\langle X,\cal{T}\hspace{.06 in} \rangle$ is completely regular?
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