The Book of Mormon’s extensive borrowings from the Bible prove that it is a modern book, since no ancient scripture could be expected to quote from so many different Bible books.
This a priori assumption is without merit. The New Testament quotes extensively from the Old Testament, while some Old Testament prophets quote from their predecessors (e.g., Jeremiah from Isaiah, Isaiah from Hosea, Obadiah from Jeremiah, etc.). The genealogy lists of Genesis 5, 10-11, 36 are repeated in the early chapters of 1 Chronicles. Isaiah 36-39 repeats material already found in 2 Kings 18-20. Much of the history found in the books of Samuel and Kings is repeated in the books of Chronicles. Nephi used most of the Isaiah quotes (those quoted in 2 Nephi 12-14) as a vehicle to explain the meaning of his own revelations from God. He could not have done this as effectively had he not quoted them for his readers. Abinadi and Jesus, during his visit to the Nephites, also used Isaiah quotes as background for explanations (in the surrounding chapters) of doctrinal matters. In the New Testament, the apostle Paul quotes extensively from Old Testament and ancient Greek sources, sometimes without crediting the original author.