Instead of God’s law being outside of them, the Lord would write it on their heart that is their mind and will. After those days” means after Israel’s return from exile. “Writing on the heart” suggests a contrast with the Old Mosaic Covenant, which was written on stone tablets. This replaces the 10 commandments with a changed heart, and replaces priests as human mediators with direct access to God. Having the Lord’s Word in the heart prevents sin and encourages obedience. David compared having the law written on his heart with desiring to do God’s will (Ps. 40:8) “It will become part of the nature of God’s people; it will be something done not because we have to but because we want to. The essential nature of the new covenant is God’s gift of a new heart. This change comes about through the presence of the indwelling Holy Spirit, although that is not mentioned here.
But this is the new covenant I will make with the people of Israel after those days,” says the LORD. “I will put my instructions deep within them, and I will write them on their hearts. I will be their God, and they will be my people. (Jeremiah 31:33)