John Chrysostom
If there must be another priest, or rather another priesthood, then there must also be another law. This is said against those who ask: What need was there of a new covenant?
He could have brought testimonies from prophecy: You are the sons of the prophets and of the covenant which God made with your fathers (Acts 3:25). But for now he reasons about the priesthood.
See how he wished to instill this. By saying according to the order of Melchizedek, he rejected the order of Aaron. He would not have said according to the order of Melchizedek if Aaron's order had been better.
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And when another priesthood is introduced, there must also be another covenant. It is impossible for there to be a priest without a covenant, laws, and ordinances, or for one who has received another priesthood to be guided by the former covenant.
Here an objection arose: how can one be a priest without being a Levite? But the Apostle had prepared the answer to this in what came before. Now he no longer offers the resolution, but speaks of it in passing.
I have said, he says, that the priesthood has been changed. Therefore the covenant also must be changed. It is changed not only in the manner of actions and ordinances, but also in the tribe, since it had to be changed in the tribe as well.
How? By the change of the priesthood, he says. That is, it passed from tribe to tribe, from the priestly tribe to the royal, so that the royal and the priestly might become one.
Notice the mystery. First there was the royal tribe, and afterward the priestly. So also in Christ, Who was always King, and became Priest when He assumed flesh, when He offered sacrifice.
Do you see the change? What was a point of objection he presents as a necessary consequence of the events...