This proves the blood represents the testimony, or testament , aka covenant.
Exodus 24:8
And Moses took the blood, and sprinkled it on the people, and said, Behold the blood of the covenant, which the Lord hath made with you concerning all these words.
Hebrews 9:1-20
King James Version
9 Then verily the first covenant had also ordinances of divine service, and a worldly sanctuary.
2 For there was a tabernacle made; the first, wherein was the candlestick, and the table, and the shewbread; which is called the sanctuary.
3 And after the second veil, the tabernacle which is called the Holiest of all;
4 Which had the golden censer, and the ark of the covenant overlaid round about with gold, wherein was the golden pot that had manna, and Aaron's rod that budded, and the tables of the covenant;
5 And over it the cherubims of glory shadowing the mercyseat; of which we cannot now speak particularly.
6 Now when these things were thus ordained, the priests went always into the first tabernacle, accomplishing the service of God.
7 But into the second went the high priest alone once every year, not without blood, which he offered for himself, and for the errors of the people:
8 The Holy Ghost this signifying, that the way into the holiest of all was not yet made manifest, while as the first tabernacle was yet standing:
9 Which was a figure for the time then present, in which were offered both gifts and sacrifices, that could not make him that did the service perfect, as pertaining to the conscience;
10 Which stood only in meats and drinks, and divers washings, and carnal ordinances, imposed on them until the time of reformation.
11 But Christ being come an high priest of good things to come, by a greater and more perfect tabernacle, not made with hands, that is to say, not of this building;
12 Neither by the blood of goats and calves, but by his own blood he entered in once into the holy place, having obtained eternal redemption for us.
13 For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh:
14 How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God?
15 And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.
16 For where a testament is, there must also of necessity be the death of the testator.
17 For a testament is of force after men are dead: otherwise it is of no strength at all while the testator liveth.
18 Whereupon neither the first testament was dedicated without blood.
19 For when Moses had spoken every precept to all the people according to the law, he took the blood of calves and of goats, with water, and scarlet wool, and hyssop, and sprinkled both the book, and all the people,
20 Saying, This is the blood of the testament which God hath enjoined unto you.”
Notice verse 9 and 14 talk about purifying and purging our conscience from evil.
This is where the Holy Spirit comes in and writes Gods words on our hearts and minds. This is the New Covenant / Testament God spoke of in the OT prophets, where he would write His words and laws on the heart of the people.
This is what Paul is showing here….
2 Corinthians 3:3
Forasmuch as ye are manifestly declared to be the epistle of Christ ministered by us, written not with ink, but with the Spirit of the living God; not in tables of stone, but in fleshy tables of the heart.
