Jamieson, Fausset and Brown on Hebrews 10:15

15. The Greek, has "moreover," or "now."

is a witness--of the truth which I am setting forth. The Father's witness is given He 5:10. The Son's, He 10:5. Now is added that of the Holy Spirit, called accordingly "the Spirit of grace," He 10:29. The testimony of all Three leads to the same conclusion (He 10:18).

for after that he had said before--The conclusion to the sentence is in He 10:17, "After He had said before, This is the covenant that I will make with them (with the house of Israel, He 8:10; here extended to the spiritual Israel) ... saith the Lord; I will put (literally, 'giving,' referring to the giving of the law; not now as then, giving into the hands, but giving) My laws into their hearts ('mind,' He 8:10) and in their minds ('hearts,' He 8:10); I will inscribe (so the Greek) them (here He omits the addition quoted in He 8:10, 11, I will be to them a God ... and they shall not teach every man his neighbor ...), and (that is, after He had said the foregoing, He then adds) their sins ... will I remember no more." The great object of the quotation here is to prove that, there being in the Gospel covenant, "REMISSION of sins" (He 10:17), there is no more need of a sacrifice for sins. The object of the same quotation in He 8:8-13 is to show that, there being a "NEW covenant," the old is antiquated.