The addition of this phrase appears to be influenced by the New Testament passage that joins the unity of the Father and Son with the Holy Ghost as the one to bear record of the divinity of the Father:
1 John 5:7
7 For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.
The alteration in the 3 Nephi context is that the Holy Ghost is given as the witness of the Father and the Son, rather than a triune witness indicated in 1 John.