The law of Moses was abolished at the cross. Christians are now under the law of Christ.
The phrase "law of Christ" is only found in two places in the Bible and both were written by Paul.
What did Paul mean when he used this phrase?
To the Jews I became as a Jew, so that I might win Jews; to those who are under the Law, as under the Law though not being myself under the Law, so that I might win those who are under the Law; to those who are without law, as without law, though not being without the law of God but under the law of Christ, so that I might win those who are without law. (1 Corinthians 9:20-21)
Paul states that he is approaching those who are "without the law" (i.e. Gentiles) from the perspective of being "without law" but clarifies his statement to avoid any confusion: "though not being without the law of God" but "under the law of Christ".
Not without the law of God = Paul has the law of God.
Under the law of Christ = Paul has the law of Christ.
Is Paul attempting to contrast "the law of God" with "the law of Christ"? Not at all! Paul is affirming they are one and the same: the law of God is the law of Christ.
The other time Paul uses this phrase is in Galatians:
Bear one another's burdens, and thereby fulfill the law of Christ. (Galatians 6:2)
Nothing in this passage of Galatians provides an indication that the law of Christ is anything other than the law of God.
Most Christians will agree that they are "under the law of Christ"... they just don't recognize what it is.