Brigham teaches the Saints that what makes one the President of the Church is not being a Prophet but instead their holding the Keys of the Priesthood.

Speech / Court Transcript
Brigham Young
Scribed Verbatim

Brigham Young, "The Temple Corner Stones—the Apostleship, &c." Journal of Discourses, 26 vols. (Liverpool: F.D. and S.W. Richards, 1854), 1:133

F. D. and S. W. Richards
Brigham Young
Latter-day Saints, Reading Public

Perhaps it may make some of you stumble, were I to ask you a question. Does a man's being a Prophet in this Church prove that he shall be the President of it? I answer, No! A man may be a Prophet, Seer, and Revelator, and it may have nothing to do with his being the President of the Church. Suffice to say, that Joseph was the President of the Church, as long as he lived; the people chose to have it so. He always filled that responsible station by the voice of the people. Can you find any revelation appointing the President of the Church? The keys of the Priesthood were committed to Joseph, to build up the Kingdom of God on the earth, and were not to be taken from him in time or in eternity, but when he was called to preside over the Church, it was by the voice of the people; though he held the keys of the Priesthood, independent of their voice.

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