In 2 Chronicles 19:11 and Malachi 2:7 there is evidence of the living continuing authority of the Mosaic priesthood. Isaiah in Chapter 22:20-25 makes a very interesting prophecy. His oracle states there will be an abolition of the Old Testament high priests and the establishment of a new priesthood of the House of David. This priesthood is very specific however.
On that day I will summon my servant Eliakim (means loyal to God), son of Hilkiah; I will clothe him with your robe, and gird him with your sash, and give over to him your authority. He shall be a father to the inhabitants of Jerusalem, and to the house of Judah. I will place the key of the House of David on his shoulder; when he opens, no one shall shut, when he shuts, no one shall open.......etc.
Very interesting. It sounds like a passage in the New Testament.
Matthew 16:18-19 I for my part declare to you, you are 'Rock', and on this rock I will build my church, and the jaws of death shall not prevail against it. I will entrust to you the keys of the kingdom of heaven. Whatever you declare bound on earth shall be bound in heaven; whatever you declare loosed on earth shall be loosed in heaven."
But, none of this went into effect before Christ's death. In fact, when Jesus admonishes the scribes and Pharisees for their hypocrisy he still acknowledges their priestly authority (Matthew 23:1-3). The Old Testament sacrifice, ritual and law was in effect until it was replaced by the NT Sacrifice on the Cross. With the New Covenant came a new priesthood. The Commission of the Apostles in Matthew 28 and Pentecost began their priesthood (also Mark 16, Luke 24:44-49,John 20:22-23, Acts 2).
In Acts Chapter 1 there is direct mention of the special apostolic office and ministry which has been established: May another take his office and again in verses 24-25, Make known to us which of these two you choose for this apostolic ministry, replacing Judas, who deserted the cause and went the way he was destined to go. This directly says the Apostles knew their position was an office and set apart from other believers.
In Acts 6 they recognize the need for assistants due to the growth of the Christian community. This is the first example (of many) I can find of the Apostolic transmission of Holy Orders to earthly men: They presented these men to the apostles, who first prayed over them and then imposed hands on them. The word of God continued to spread, while at the same time the number of the disciples in Jerusalem enormously increased. There were many priests among those who embraced the faith.
Acts also makes it clear that Peter and the apostles no longer submit to the authority of the Jewish Sanhedrin.