Charles Hodge, in speaking on the aftermath of the Council of Nicaea (325 AD) and the Arian Resurgency, touches upon a flaw in the Roman Catholic definition of what the church is, and how it is identified.
In dealing within this undeniable fact, Romanists and Romanizers are forced to abandon their principle. Their doctrine is that the external Church cannot err, that the majority of the bishops living at any one time cannot fail to teach the truth. But under the reign of the Emperor Constantius, it is undeniable that the vast majority, including the Bishop of Rome, did renounce the truth. But, says Bellarmin, the Church continued and was conspicuous in Athanasius, Hilary, Eusebius, and others. And Mr. Palmer, of Oxford says, “The truth was preserved under even Arian bishops.” But the question is not, whether the truth shall be preserved and confessed by the true children of God, but, whether any external, organized body, and specially the Church of Rome, can err in its teaching. Romanists cannot be allowed, merely to meet an emergency, to avail themselves of the Protestant doctrine that the Church may consist of scattered believers. It is true as Jerome teaches in the passage above quoted, “Ubi fides vera est, ibi Ecclesia est.” But that is our doctrine, and not the doctrine of Rome. Protestants say with full confidence, “Ecclesia manet et manebit.” But whether in conspicuous glory as in the time of David, or in scattered believers as in the days of Elias, is not essential. [source]