on what basis did the bishop of rome claim authority over other bishops
Because of its association with the supposed position of Peter among the Apostles, the function that, within the Roman Catholic Church, is exercised by the Bishop of Rome among the bishops as a whole is referred to as the Petrine function, and is generally believed to be of divine institution, in the sense that the historical and sociological factors that influenced its development are seen as guided by the Holy Spirit. Not all Roman Catholic theologians see a special providential providence as responsible for the result, but most see the papacy, regardless of its origin, as now essential to the Church's structure.