B737 Captain reported two braking systems anomalies during approach. Flight executed a missed approach and returned for a normal landing.

Narrative

While on the RNAV to XX in ZZZ approaching the final turn on base with gear down and flaps 15; the Auto Brake Disarm light and the Anti Skid Inop lights both illuminated. When selected off then back on; they both re-illuminated. An uneventful go around was initiated. We were then given vectors around ZZZ while we ran the Auto Brake Disarm QRC followed by the Anti Skid Inop QRC. With the Anti Skid system being more restrictive; we continued the flight operating under that checklist and followed that through landing. When checklists were completed; we contacted Dispatch who also phone patched in Maintenance Control. After describing what we saw; we agreed that a landing in ZZZ would be safe; and the landing distance numbers agreed with the same numbers both the FO and I have independently calculated. The winds were favoring XX; however; in order to get a longer runway; we requested one of the XY's. Priority handling was requested and we chose XYR for the longer landing distance. Flight attendants were then briefed followed by the passengers. A normal approach and landing occurred following the landing procedure outlined in the QRH. A normal taxi in and deplaning at the gate followed. Maintenance staff met us in the flight deck and we debriefed what we had experienced as far as the warning lights as well as what QRH procedures we had accomplished. Phone calls were then made to Dispatch and the Chief Pilot.