1It is reported commonly that there is fornication among you, and such fornication as is not so much as named among the Gentiles, that one should have his father's wife.
2And ye are puffed up, and have not rather mourned, that he that hath done this deed might be taken away from among you.
3For I verily, as absent in body, but present in spirit, have judged already, as though I were present, concerning him that hath so done this deed,
4In the name of our Lord Jesus Christ, when ye are gathered together, and my spirit, with the power of our Lord Jesus Christ,
5To deliver such an one unto Satan for the destruction of the flesh, that the spirit may be saved in the day of the Lord Jesus.
First, Paul wasn't in Corinth to witness the sin in Corinth himself. You can see in v. 1 that it was reported to him by others. We can and should believe reports from others. If several people are saying that they are seeing the same thing, that report can be believed. And Paul believed. An unwillingness to believe a report is just that, an unwillingness to believe it. It should be believed.
Second, according to v. 3, Paul believed that he could unilaterally decide, based upon that report, without even being present, that discipline should have been enacted. And we are talking about the church at Corinth here, who was harboring a man in the state of committing incest without repentance. Could anyone there be trustworthy to give an accurate report when no one had stepped up to the discipline of this man? Paul had already decided that this man should have been removed from the church.
Third, in v. 4, Paul calls upon the vertical authority of Jesus Himself in the case of discipline. Jesus is acting in church discipline. The "power of our Lord Jesus Christ" goes back to Jesus' teaching on discipline in Matthew 18. What is bound or loosed on earth is bound or loosed in heaven. Heaven is acting in a case of church discipline, even if it were the church at Corinth. And then when two or three witnesses speak in a case of church discipline, Jesus' presence is there in their witness.
Why would there need to be so much of an introduction to the method of church discipline, calling upon the authority of Jesus Christ in it? Because men are going to question and attack church discipline. They will have their "reasons" to do so, moving into the credibility and the veracity of the church. But they are crossing Jesus when they do so, because Paul said that Jesus' power was involved as well. Paul calls upon the "name of Jesus Christ" because church discipline is exactly what Jesus would have done if He were there. We get a taste of that from Revelation 2:18ff when Jesus speaks to the church at Thyatira for harboring a Jezebel in its midst.
Fourth, Paul relates the authority of the church in this matter, when he writes, "when ye are gathered together" in v. 4. There was no authority over the church of Corinth in its discipline of this member. Paul says "ye." That church was operating with the power of Jesus Christ when it gathered together. A person or other church which ignores the discipline by a church, when the issue is witnessed by two, three, or even commonly reported, they disrespect Jesus. It's His power by which this is done. Surely the credibility of the church of Corinth could be questioned, but even with that church, it was the power of Jesus at work in discipline.
A separate church, gathered together, has the authority to discipline a member. That is the kind of authority God gives a church. And God trusts a church in doing that. It is a shame when other professing Christians or churches will not do that. Really it is more than a shame, it is a repudiation of Jesus Christ Himself, because the act of discipline was done by His authority. The discipline of a single church is the act of Jesus. His authority rests with a church.
Some might judge Bethel Baptist Church in El Sobrante, CA to be illegitimate simply because it believes in one text of Scripture, which English translation is only the King James Version, or because it believes God expects men and women to wear clothes with designed distinctions between genders. Or they might judge that Bethel Baptist Church should be ignored in its discipline because its pastor, like other pastors, makes strong, dogmatic statements about belief as if scripture is perspicuous. They might feel justified in disbelieving a violation which is commonly reported by that church. I would ask anyone like that to consider the problems of the church of Corinth, and that Paul believed that church could and should practice discipline of its members even without his presence. And then understand that you do not just oppose Bethel Baptist Church, but also the Lord Jesus Christ Himself, when you disrespect its discipline.