This lesson is a study of Heb 8.  Jesus has a more excellent ministry, Heb 8:6, than the priests after the order of Aaron because:

He ministers in a better tabernacle, v.1-2.  Jesus is a minister in the true tabernacle.  He’s seated at the right hand of God in the heavens. Heb 9:11. The high priests were ministers in the worldly sanctuary, v.5.  

He offers better gifts and sacrifices, v.3-5. Their gifts and sacrifices, were blood of bulls and goats, Heb 9:12-14.  Jesus’s gifts and sacrifices were his own body and blood.  There would be no reason for Jesus to be a high priest down here, v.4.  That’s why he wasn’t one in his earthy ministry.  In v.5 the pattern for the furniture in the tabernacle is seen in Ex 25:40.  In Rev 11:19 the heavenly temple is seen.

He is the mediator of a better covenant, v.6-13.  This is the covenant of Jer 31:31-34.  And Jer 31:35-37 was written to show us that this new covenant is still effective.  Compared to the first covenant with Moses, the new covenant has:

Better promises, v.6.  The promises of the first covenant are summarized in Deut 6:24-25.  They were to do the statutes and thereby live.  In the second covenant, v.10, the laws will be put into their mind, and he will write them in their hearts.

No Fault. The first had a fault, v.7-8.  The law couldn’t make them perfect, Heb 7:19; 10:1.  The second is faultless, v.12.  “Their sins and their inquiries will I remember no more”.  In Heb 10:14 the are perfected.  In Rom 11:25-27 all Israel shall be saved.

No discontinuance. In the first, they continued not in my covenant, and I regarded them not, v.9.  In the second, v.10-11, he will be their God and they shall be his people.  All shall know him from the least to the greatest.

No need to teach. In the first, they had to teach the people. v.11. In the second, they do not have to teach them.  In Lev 10:9-11 the priests were instructed to teach them.  In Deut 4:6-10, the parents were instructed to teach them, as well.

No remembrance of sins. In the first, there is remembrance of sins, Heb 10:3.  In the second, their sins “will I remember no more”, v.12.

In v.13, we call the first covenant Old (as in Old Testament), because God called the second covenant New (as in New Testament).  And the old is ready to vanish away, meaning that it will not be gone at this point.  They will be sacrificing at temple, Rev 11, and they can’t flee on sabbath, Matt 24:20.  The law is still a shadow of things to come, Col 2:17.  

To study the prior lesson, see After The Order Of Melchisedec. To study the next lesson, see The More Perfect Tabernacle.