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The promise of Psalm 110 that God was going to appoint a perpetual high priest after the order of Melchizedek rather than Levi (highlighted in the previous chapter) implied the ultimate abolishment of the Levitical priesthood. This, in turn, implied that a significant part of the Mosaic Law that had anything to do with the Levitical priesthood would become obsolete. Thus one could easily see how the arrival of the promised perpetual high priest would necessitate a complete changing of the Law of Moses. We read in 7:12: "For when the priesthood is changed, of necessity there takes place a change of law also." That change was nothing less than the end of the old covenant and the beginning of the new. Jews who were (and are) in sync with God's plan have made that transition.
This reality becomes even more obvious when we realize that the priestly ministry under the old covenant was simply a foreshadowing of Christ's priestly ministry. In this chapter of Hebrews, the author declares that the earthly tabernacle, for which Moses was given detailed construction plans by God, was a copy of a tabernacle in heaven (8:5; 9:23-24). Jesus was "a minister in the sanctuary and in the true tabernacle, which the Lord pitched, not man" (8:2). Thus, not only is Jesus personally superior in His priestly office as compared to any former priest, but He also has a superior ministry to any former priests as well. The high priests of the old covenant performed their ministry in an earthly, man-made tabernacle. They stood before a symbol of God's throne in the earthly Holy of Holies to present the blood of animals. Jesus, however, presented His own blood before God's throne in the heavenly Holy of Holies.
And it was that superior act in a superior place by a superior priest---foreshadowed thousands of times by inferior acts in inferior places by inferior priests---that inaugurated a superior new covenant. That superior new covenant, just like that superior high priest, had also been promised in the Old Testament. The author of Hebrews cites such a promise from Jeremiah 31:33-34, arguing that "if that first covenant had been faultless, there would have been no occasion sought for a second" (8:7). Because God promised a second covenant, that implied the fact that He found fault with the first, and naturally anyone who remained under the old covenant when the new covenant was inaugurated would be missing the mark.
The new covenant has made the old covenant obsolete (8:13). And we are now obligated to obey the law of Christ and not the Law of Moses. Any Christian, Jew or Gentile, who is trying to live under the old covenant is going backwards 2,000 years with God. Those who teach that Christians are obligated to keep the Mosaic Law might just as well start sacrificing animals to receive forgiveness of their sins!
Hundreds of years before the inauguration of the new covenant, God foretold through Jeremiah a few of the benefits that would be enjoyed by those who would partake of its promises. It would be superior to the old covenant because God would write His laws on hearts and minds, an inward work that would result in holy lives. Everyone who experienced that inward work would "know the Lord" (8:11), because knowing Him is revealed by a lifestyle of obedience to Him. So we see that hundreds of years before Jesus' incarnation, God was already revealing the true nature of our salvation and exposing the false grace that is so often being peddled today, a grace that doesn't change anyone's behavior.
The apostle John similarly wrote, "By this we know that we have come to know Him, if we keep His commandments" (1 John 2:3). Knowing the Lord is synonymous with following His commandments, which was true even under the old covenant. The Lord also once said through Jeremiah:
Did not your father eat and drink and do justice and righteousness? Then it was well with him. He pled the cause of the afflicted and needy; then it was well. Is not that what it means to know Me? declares the Lord (Jer. 22:15-16).