Global and local choice functions

We prove, by an elementary reflection method, without the use of forcing, that ZFGC (ZF with a global choice function) is a conservative extension of ZFC and that every model of ZFC whose ordinals are cofinal (from the outside) with ω can be expanded to a model of ZFGC (without adding new members). The results are then generalized to various weaker forms of the axiom of choice which have global versions.