Moral law, also called God’s law or Christ’s law has always existed because murder, adultery, stealing and things like these always have been and always will be wrong. The Covenant God Gave to Abraham was a promise that he and all of us who believe on and trust the Messiah for salvation will be saved. In Galatians. 3:24, the law of Moses was added later to show what perfection would be if we could attain it and to show us we all need a savior.
James 2:8 If ye fulfil the royal law according to the scripture, Thou shalt love thy neighbour as thyself, ye do well: (KJV)

The Abrahamic Covenant
Some say Christians are under a new, completely different covenant; and the covenant of Moses has nothing to do with us. That means whatever is not directly stated in the New Testament, as being still in force is null and void.

Some will argue that we are not under a completely different covenant but simply the old one slightly changed by grace. That means whatever is not directly stated in the New Testament as being done away with is still in force. They say only the Ceremonial part of the law of Moses has been done away, then they argue about what laws are ceremonial, civil, or moral. For example if the dietary laws were for health reasons then they are still in force, but if they are to teach us about sin and holiness then they are ceremonial, and therefore not in effect.

There is no scripture that says that only the ceremonial laws of Moses are done away.
It is not a valid argument to say "since it is a perpetual covenant lasting throughout all generations, we must be under it." Inside a covenant many things may change from time to time. e.g. the Tabernacle to the temple, the firstborn to the Levites ect. The parts that are perpetual will never change while that covenant is still standing. However, when the covenant ends all those things of that covenant both perpetual and not, will end also. One example of this is the Sabbath, in Exodus 31:16 it continues forever, but in Romans 14:5-8 Paul says Sabbath observance does not matter any longer and in Colossians. 2:16-17 he says not to let anyone judge you in whether or not you observe the Sabbath.

In Acts 15:24-29 James and all the Jews agreed that Gentile Christians should not be circumcised or keep any of the law of Moses, only stay away from Idols and their rituals, and fornication.

Lets look at what Paul said about the old covenant of the Mosaic law and the new covenant of Grace.
Hebrews 7:18-19 There is, on the one hand, the abrogation of an earlier commandment because it was weak and ineffectual 19 (for the law made nothing perfect);
Hebrews 8:6-7,13 But Jesus has now obtained a more excellent ministry, and to that degree he is the mediator of a better covenant, which has been enacted through better promises. 7 For if that first covenant had been faultless, there would have been no need to look for a second one. 13 In speaking of "a new covenant," he has made the first one obsolete. And what is obsolete and growing old will soon disappear.
Hebrews 9:15-16 For this reason he is the mediator of a new covenant, so that those who are called may receive the promised eternal inheritance, because a death has occurred that redeems them from the transgressions under the first covenant. 16 Where a will is involved, the death of the one who made it must be established.
Hebrews 10:9b-10 He abolishes the first in order to establish the second. 10 And it is by God’s will that we have been sanctified through the offering of the body of Jesus Christ once for all. (NRSV)

The Promise
Galatians 3:15-19 Brothers and sisters, I give an example from daily life: once a person's will has been ratified, no one adds to it or annuls it. ...17 My point is this: the law, which came four hundred thirty years later, does not annul a covenant previously ratified by God, so as to nullify the promise. 18 For if the inheritance comes from the law, it no longer comes from the promise; but God granted it to Abraham through the promise. 19 Why then the law? It was added because of transgressions, until the offspring would come to whom the promise had been made; and it was ordained through angels by a mediator. (NRSV)
Do you wish to serve Angels or Christ?

The Covenant of Grace is like a country that in 1970 passed a law stating all who murder or steal be put into prison for life. Then in 1980 a law was passed that stated all who murder or steal or don’t keep the Sabbath get the death penalty. In 1990 the 1980 law was repealed. During a Friday night party two people, who were having a good time and talking about how they didn’t have to light the Sabbath candles, got into a debate on what would happen if one of them stole a car. One said they would get the death penalty because only the ceremonial law of Sabbath lighting was repealed. The other said "no the entire law was repealed so I would get away with it." To prove his point he steals a car right in front of a police officer. To their surprise he did not get away with it nor get the death penalty but was instead put into prison for life. Yes, the whole Mosaic law was repealed, but that does not change the preexisting moral law.

The covenant of grace that Jesus brought is a fulfillment of God’s promise to Abraham.
Galatians 2:17-21 But if, in our effort to be justified in Christ, we ourselves have been found to be sinners, is Christ then a servant of sin? Certainly not! 18 But if I build up again the very things that I once tore down, then I demonstrate that I am a transgressor. 19 For through the law I died to the law, so that I might live to God. I have been crucified with Christ; 20 and it is no longer I who live, but it is Christ who lives in me. And the life I now live in the flesh I live by faith in the Son of God, who loved me and gave himself for me. 21 I do not nullify the grace of God; for if justification comes through the law, then Christ died for nothing. (NRSV)

In verse 17 if we obey the Great Commission, "go and make disciples of all nations," we break the law of Moses by eating with and going in the houses of gentiles. Paul is saying that since we can’t do both, no matter what we do, we are found to be sinners by someone’s standard. Christ is not making us sin since we are not under the old covenant. In verses 19 and 21 if we are not dead to the law (separated from it and ignoring it) we nullify the grace of God.

Galatians 4:1-3 What I am saying is that as long as the heir is a child, he is no different from a slave, although he owns the whole estate. 2 He is subject to guardians and trustees until the time set by his father. 3 So also, when we were children, we were in slavery under the basic principles of the world. 4 But when the time had fully come, God sent his Son, born of a woman, born under law, 5 to redeem those under law, that we might receive the full rights of sons. (NIV)

The Romans had what they called an adoption ceremony, which was a type of graduation. Paul is saying that Children in a boarding school even though they are heirs to a large company, have to go through a lot of rules to train them. Once they have graduated they are no longer subject to the rules. The point was that while they were under those rules, hopefully they learned who they really are, self control and that every thing they have was given them by God.

Perpetuity of which Law?
Whether or not the new covenant is a completely different one or the old one with some changes is irrelevant. Paul’s teaching about our liberty in Christ is all too clear. Paul states we are to obey the Law of Christ not the Law of Moses.

Christ’s law is:
Matthew 22:37-40 He said to him, " ‘You shall love the Lord your God with all your heart, and with all your soul, and with all your mind.’ 38 This is the greatest and first commandment. 39 And a second is like it: ‘You shall love your neighbor as yourself.’ 40 On these two commandments hang all the law and the prophets." (NRSV)
Galatians 6:2 Bear ye one another’s burdens, and so fulfil the law of Christ. (KJV)
1 Corinthians 9:20-21 To the Jews I became as a Jew, in order to win Jews. To those under the law I became as one under the law (though I myself am not under the law) so that I might win those under the law. 21 To those outside the law I became as one outside the law (though I am not free from God’s law but am under Christ’s law) so that I might win those outside the law. (NRSV)

Rom 10:4 Christ is the end of the law so that there may be righteousness for everyone who believes. (NIV)
Romans 13:9 The commandments, "Do not commit adultery," "Do not murder," "Do not steal," "Do not covet," and whatever other commandment there may be, (still in force) are summed up in this one rule: "Love your neighbor as yourself."
Paul here says any moral law that is still in force is because it is part of the Law of Christ.

2 Corinthians 3:7-11 Now if the ministry of death, chiseled in letters on stone tablets, came in glory so that the people of Israel could not gaze at Moses’ face because of the glory of his face, a glory now set aside, 8 how much more will the ministry of the Spirit come in glory? 11 for if what was set aside came through glory, much more has the permanent come in glory! (NRSV)
Here in verse 11 Paul says that all the law of Moses, even the 10 commandments are set aside!

These verses show that not just the ceremonial laws are gone:
Ephesians 2:15 Having abolished in his flesh the enmity, even the law of commandments contained in ordinances; for to make in himself of twain one new man, so making peace; (KJV)
Romans 7:4 Wherefore, my brethren, ye also are become dead to the law by the body of Christ; that ye should be married to another, even to him who is raised from the dead, that we should bring forth fruit unto God. (KJV)
1 Timothy 1:9 Knowing this, that the law is not made for a righteous man, but for the lawless and disobedient, for the ungodly and for sinners,... (KJV)
Galatians 3:19 Why then the law? It was added because of transgressions, until the offspring would come to whom the promise had been made; and it was ordained through angels by a mediator. (NRSV)
Galatians 6:13 Even the circumcised do not themselves obey the law, but they want you to be circumcised so that they may boast about your flesh. (NRSV)
Galatians 5:1 It is for freedom that Christ has set us free. Stand firm, then, and do not let yourselves be burdened again by a yoke of slavery. 4 You who are trying to be justified by law have been alienated from Christ; you have fallen away from grace. 13 You, my brothers, were called to be free. But do not use your freedom to indulge the sinful nature; rather, serve one another in love. (NIV)

If this is ceremonial law only, how could it be used to indulge sin?
Antinomianism is a term meaning "no laws at all" or anarchy. It comes from "anti" meaning against and "nomos" meaning law. Paul never meant that there are no laws he always insisted that we must be moral.

1 John 3:9 Those who have been born of God do not sin, because God's seed abides in them; they cannot sin, because they have been born of God. (NRSV)
Romans 14:13 Therefore let us stop passing judgment on one another. Instead, make up your mind not to put any stumbling block or obstacle in your brother's way. (NIV)
1 Corinthians 8:9,12 Be careful, however, that the exercise of your freedom does not become a stumbling block to the weak. 12 When you sin against your brothers in this way and wound their weak conscience, you sin against Christ. (NIV)

Why then did Jesus say he came not to abolish the law but to fulfill it?
There are two Greek words translated abolish. Katargeo means to indirectly cause something to be destroyed and kataluo meaning to directly destroy something. In Matthew 5:17, Jesus used kataluo. He did not come to have us all disobey the Law we were under. He came to destroy Sin. After that we are under a new law. Therefore the Mosaic Law, not destroyed, is useless to us because we have nothing to do with it.

In conclusion we can state that as Christians we are obligated to do what is right, obey Christ and the New Testament teachings. We should study the old covenant of Moses because it was "written for our learning." Romans 15:4 but we should not confuse the two. The old covenant of Moses was written for the Jews in time past and was completely dissolved when Christ died on the cross.