How do Evangelicals explain when Paul writes "I, not the Lord"? Paul writes in 1 Corinthians 7:10 (NKJV):


  Now to the married I command, yet not I but the Lord: A wife is not to depart from her husband 


In verse 12 Paul writes:


  But to the rest I, not the Lord, say: If any brother has a wife who does not believe, and she is willing to live with him, let him not divorce her. 


How do Evangelicals explain this in line with scripture inspired by God?
Evangelical commentators generally hold that Paul is speaking authoritatively in 1 Corinthians 7:12, and only indicating that he was not aware of any direct teaching on this subject by Jesus during his earthly ministry.  The Reformation Study Bible explains:

With regard to the problem treated in vv. 10, 11, there was a well-known instruction given by Jesus during His earthly ministry (Mark 10:1–12).  In vv. 12–16, however, Paul discusses a difficult situation that had not been expressly addressed by the Lord while He walked the earth.  His apostolic commands, however, come by inspiration and have divine authority, as 14:37 makes clear.

Others who make this point include The New Bible Commentary:

The problem of mixed marriages would hardly have arisen during Christ's ministry, and so Paul cannot quote a specific word of the Lord.  But his own judgment is given authoritatively.

And Barry G. Webb, writing in The Enduring Authority of the Christian Scriptures, edited by D. A. Carson, says:

Paul's distinction between what he gives as a command of Christ (&quot;not I, but the Lord&quot;) and what he gives only as his own word (&quot;I, not the Lord&quot;) cannot be taken as distinguishing between what has apostolic authority and what does not.  He writes the entire epistle as &quot;an apostle of Christ Jesus by the will of God&quot; (1:1).  The basic distinction seems to be between appealing to a specific teaching of Jesus, and speaking as an apostle without being able to appear to any specific teaching of Jesus regarding the issue at hand. (chapter 19, fn. 17)

The ESV Study Bible says:

Paul views his admonition here as authoritative and inspired, not merely as human wisdom (v. 40; cf. 14:37–38).

The MacArthur Study Bible:

Not a denial of inspiration or an indication that Paul is giving human opinion, but simply a way of saying that Jesus had not spoken on this and God had not previously given revelation on the matter, as Paul was then writing.

Constable's Commentary agrees:

For mixed couples, Paul could not cite a teaching of Jesus, because He had
not spoken on this subject. At least, as far as Paul knew He had not.
Nevertheless, the risen Lord inspired Paul's instructions on this subject, so
they were every bit as authoritative as the teaching Jesus gave during His
earthly ministry.

And so does the Moody Bible Commentary:

Paul was not aware of any teaching on mixed marriages given by Jesus, but as an apostle inspired by God, Paul's words are binding.

Conclusion
Evangelicals broadly agree that what Paul writes in 7:12 is inspired and just as authoritative as the other teachings that Paul gives.
That said, it's perhaps worth noting that Evangelicals accept that some apostolic speeches and writings are not strictly inerrant in their content (though the record is still accurate).  For example, they admit that Paul speaks erroneously in Acts 20:25, and that in 2 Corinthians 8, he gives advice, not a divine command.  On the latter case, Barry Webb writes:

Just as the Old Testament Scriptures contain &quot;command&quot; (law) and &quot;advice&quot; (wisdom teaching), so do the writings of the apostles; but all of it is Scripture.

But with respect to the verses in question here, these comments are merely tangential – Evangelicals generally accept 1 Corinthians 7:10ff. as authoritative commands, under the inspiration of God.