Many people do believe that the statement "If the Book of
Mormon is true then The Church of Jesus Christ of Latter-day Saints is the
true church" is a true statement. I don't believe that is true--if the Book
of Mormon were true, that would be further evidence that the LDS Church is
false.
The beliefs of the church were in harmony with the Book of
Mormon in the early 1830's, but since then the church has been continuously
evolving and mutating, and its beliefs are now only loosely correlated with
the Book of Mormon. If churches go into apostasy for teaching false
doctrine, then it is probable that the Mormon Church has fallen into
apostasy.
For an example of how belief in the Book of Mormon condemns
the Mormon Church, consider the story of Alma Senior. He was one of evil
King Noah's false priests. He hears Abinadi preach that Jesus is "God
himself" and other ideas that make the most sense if you accept that Abinadi
was a Trinitarian. Alma's heart is pricked, and he begins teaching in the
wilderness, and baptizes himself and then a whole bunch of people. He then
leads his people to Zarahemla where he establishes churches that are only
loosely associated with each other. These various churches were
collectively "the church of God" not because they were established by the
priesthood authority--something that was never conferred upon Alma--but
rather because "there was nothing preached in all the churches except it
were repentance and faith in God." This is the first mention of a church
being established in the Book of Mormon, and it doesn't happen until 480
years after Lehi left Jerusalem. They preached Trinitarian ideas, and the
church was established without priesthood authority (Mosiah 17-25).
The concept of churches and priesthood authority as taught
in the Book of Mormon is fundamentally different than the way it is
understood in the contemporary LDS Church. In Mormon Hierarchy: Origins
of Power, Michael Quinn demonstrates that authority in Mormonism has
mutated from being something that was possessed by anybody who was moved by
the spirit to being something that was totally controlled by the hierarchy
of the church without regard to the spirit. When you understand this,
conviction that the Book of Mormon is true would tend to condemn the LDS
Church rather than validate it. I believe that this is a true statement
regarding the implications of the Book of Mormon: "if the Book of Mormon is
true, then the true church is a splinter church such as the RLDS or
Strangites that has teachings that more closely resemble what is taught in
the Book of Mormon, or the true Church isn't upon the face of the earth, or
the whole concept of a 'true church' is a misnomer because God doesn't
establish churches."