Air carrier flight crew reported a ground conflict with another aircraft during taxi for takeoff. Flight crew brought aircraft to a stop; then continued taxi.

Narrative

At approximately XA:45 we were taxiing out of the gate area for departure to ZZZ. Ramp had told us to taxi on G; hold short of A and contact Ground. We were told to contact Ground on 121.65 but after a couple failed attempts we switched to 121.9 and reached them. Ground Control instructions were to taxi to Runway 22R turn Left on B; E; cross Runway 13L. We were never told to give way to another aircraft or that an aircraft was giving way to us. I cleared left" and the FO (First Officer) "cleared right" as we were approaching B from G. The FO then stated that he was going heads down. I looked to the right and saw an aircraft taxiing on B coming towards us. We appeared to have plenty of room to make our turn and I assumed they were going to be holding short of G for us. It also appeared to me that they were taxiing faster than usual and did not seem to be slowing down so I started slowing down. It quickly became apparent that they were not stopping and I had to quickly come to a stop just prior to B. As they passed in front of us we queried the Ground controller if we were supposed to be giving way to that aircraft to which he replied; "Oh yeah; give way to him" as they had already passed us. Had I just followed our taxi instructions and not double checked to the right as the FO was heads down we would have been in a position for a likely collision with the other aircraft. I'm not sure at the speed that they were taxiing if they would have been able to stop in time. The aircraft was a Aircraft Y; ZZZ1-JFK."