Drywall is not structural so eliminating the stud by cutting through it will affect the integrity of the wall. As noted above you will not have the normal points to mount the medicine cabinet, and attempts to use the drywall for mounting will probably fail over time. To do this correctly (IMHO) you would need to cutout the stud, add new horizontal pieces that would be connected to the existing vertical studs and to the cut stud both above and below the location of the new medicine cabinet. then you would need to add 1 or 2 new vertical studs the the horizontal ones to support the cabinet, at least this is what I would do.
good luck with your project.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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