Spirit’s work about the New Covenant?

Was Asked – Is the Spirit’s work that which is new about the New Covenant?

My Answer was – Yes! I simply do not see one can argue the fact that the work of the New Covenant is not established today! I have heard a number of times by mainly classical dispensationalist (who have made up their minds that no covenant is given to the church) that this is not in order today for the local church. I think to show how the New Covenant is consummate by the pouring of the Spirit on and in man is best to go to the Scriptures which seem at least to myself in making this clear.

The New Covenant as the Law of God in the human mind
Ezekiel 36: 26-27 – A new heart I will give you, and a new spirit I will put within you; and I will remove from your body the heart of stone and give you a heart of flesh. I will put my spirit within you, and make you follow my statutes and be careful to observe my ordinances.

The New Covenant as a gift from the Holy Spirit
Galatians 5:22-23 – By contrast, the fruit of the Spirit is love, joy, peace, patience, kindness, generosity, faithfulness, gentleness, and self-control. There is no law against such things.

The New Covenant as a fulfillment by the Spirit
Romans 8:4 – so that the just requirement of the law might be fulfilled in us, who walk not according to the flesh but according to the Spirit.