Hebrews 8:6-13

Now hath he obtained a more excellent ministry. More excellent than that of the Levitical priesthood. He serves in the heavenly sanctuary.

He is the meditator. The high priest was a mediator, interceding with God for the people. Christ is the mediator of a better covenant with its more glorious promises; viz., the Gospel.
For if that first covenant. The covenant of Sinai, the Law of Moses, the Old Testament.

Had been faultless. It was faulty and could not make men perfect.

See PNT Heb 7:18.
Behold, I will make a new covenant. By the mouth of Jeremiah (Jer 31:31-34), the Lord declared that he would make a New Covenant which would take the place of the Old Covenant. This shows that the Old did not meet the full needs of men. Not according to the covenant I made with their fathers, etc. See Jer 31:32. It shall differ from the covenant of Sinai made while the Lord was leading Israel up from Egypt.

Because they continued not in my covenant. Israel had broken the covenant, and been unfaithful.
This [is] the covenant that I will make, etc. See Jer 31:33. This shall be a principle of the new covenant.

I will put my laws in their mind. Under the new covenant the minds and hearts shall be given to the Lord and filled with his law. They shall "worship in spirit and in truth" (Joh 4:23,24).

Write them in their hearts. Instead of upon tables of stone.
And they shall not teach every man . . . saying, Know the Lord. See Jer 31:34. Under the old covenant children were born into covenant relation. As they grew up they had to be taught of the Lord, or they would never know him. As a matter of fact, many did not learn to know him.

For all shall know me, from the least to the greatest. But under the new covenant all must learn to know God and the Son whom he hath sent before they can enter the covenant. Every one must enter by faith in Christ, repentance of sin, and surrender to the will of God. Hence all members of the new covenant know God, "from the least to the greatest".
And their sins and their iniquities will I remember no more. When they enter into covenant relation through the Gospel every sin will be blotted out. A new [covenant], he hath made the first old. This language implied that the other covenant, the one made at Sinai, is the old covenant.

That which decayeth and waxeth old [is] ready to vanish away. This language implies that the old covenant is to pass away. That it has done so is shown in the next chapter.
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