'Basis' in the Bible

Now if perfection could have been attained through the Levitical priesthood for on this basis the people received the Law what further need would there be to speak of appointing another kind of priest according to the order of Melchizedek, not one according to the order of Aaron?

who was appointed to be a priest, not on the basis of a genealogical registry, but rather on the power of an indestructible life.