While the letter was sent to the church at Corinth, it was addressed to a wider audience:
to the church of God that is in Corinth, to those who are sanctified in Christ Jesus, and called to be saints, with all those in every place who call on the name of our Lord Jesus Christ, their Lord and ours. (1 Corinthians 1:2) [NET]
As some commentators note, this opening gives the letter a universal address:
With all that in every place.—Better translated, with all that call upon the name of our Lord Jesus Christ in every place, both theirs and ours. The teaching of the Epistle is thus addressed to the Church at large, which is composed of all who call upon the Lord Jesus, whether it be in Corinth (“our” country—the Apostle identifying himself with his converts) or elsewhere. This idea of the Church, put forward in the very opening of the Epistle, at once directs the reader’s mind from the narrow spirit of faction which was exhibiting itself at Corinth. The words of this verse contain a strong testimony to the worship of Christ, not only as being practised in the Apostolic Church, but as being one of the very marks of true union with the Church.
with all that in every place call upon the name of Jesus Christ our Lord. The Epistle, which dealt with so many and such weighty truths, was not to be treasured up as the peculiar heritage of the Corinthian Church, but was to be regarded as the common possession of the universal Church of Christ. Or perhaps it is better, with Olshausen, to regard the Apostle as reminding the Corinthians that they form only a part, and that but a small one, of the whole Church of Christ, a consideration which their self-satisfaction was leading them to forget.
This opening is intentional and necessary as Paul’s appeal will be for unity among believers:
I urge you, brothers and sisters, by the name of our Lord Jesus Christ, to agree together, to end your divisions, and to be united by the same mind and purpose. (1:10)
As Richard B. Hays comments:
The community’s life before God depends entirely on Jesus’s death on the cross (cf. 11:26; 15:1-3, and the Lord into whose dominion the community has been transferred in baptism is Jesus Christ alone. The church is saved and sustained only in the name of Jesus.1
This logic for unity in Corinth must apply to the church in every location. Hence the need to include all believers in the address: “…with all those in every place who call on the name of our Lord Jesus Christ, their Lord and ours.”
The letter to the Corinthians was written from Ephesus, to whom Paul also wrote about unity:
I, therefore, the prisoner for the Lord, urge you to live worthily of the calling with which you have been called, with all humility and gentleness, with patience, bearing with one another in love, making every effort to keep the unity of the Spirit in the bond of peace. There is one body and one Spirit, just as you too were called to the one hope of your calling, one Lord, one faith, one baptism, one God and Father of all, who is over all and through all and in all. (Ephesians 4:1-6)
The two letters not only same the same logic of unity as the body of Christ, they confirm an underlying reality of travel and communication between Christians in different cites. The practical consequences of this inter-relationship between Christians in different cites demands the practice of Communion in all locations. If this were not the case, Christians visiting Corinth would have no idea what the Corinthians were doing when they shared Communion. Likewise Corinthians visiting other cities would be confused by the failure not to engage in the practice of Communion.
In other words, if Paul did not teach Communion to all churches, the practice of Communion at Corinth would be a cause of division between Christians. The fundamental practices taught at each location must be the same or else Paul's teaching at one location would become a source of division within the body of Christ at another location.
1. Richard B. Hays, First Corinthians, John Knox Press, 1997, p.23