Brigham Young teaches that the Constitution will never be destroyed but, based on the teachings of Joseph Smith, will hang upon a thread.
Brigham Young, "Celebration of the Fourth of July." July 4, 1854, Journal of Discourses, 26 vols. (Liverpool: Amasa Lyman, 1854) 7:15
Will the Constitution be destroyed? No: it will be held inviolate by this people; and, as Joseph Smith said, ‘The time will come when the destiny of the nation will hang upon a single thread. At that critical juncture, this people will step forth and save it from the threatened destruction.’ It will be so.