[FOM] Axiom of Choice/(ultra)filters

[pax0@seznam.cz]

I have a question regarding the Axiom of Choice,
what is the strength of the following two theories:

(1)  ZF+{on every filter, there is a selector},

(2)  ZF+{on every ultrafilter, there is a selector} .

More precisely, are they strictly weaker then ZFC?

Here, by a selector on a set X I mean a function with the domain all nonempty sets in X and
with f(x) \in x for every member x in its domain.
Then ZFC is equivalent to ZF+{on every set, there is a selector}.

Thank you