Peter proceeded to quote from the first writing prophet to confirm what he had just stated. Moses had predicted that God would provide prophets similar to himself through whom He would make His will known to His people ( Deuteronomy 18:15-19; cf. Leviticus 23:29). As time passed, the Jews saw that this prophecy referred to one prophet in particular who would appear and who would be like Moses in other respects as well. [1] He would deliver and judge His people. Thus believers in Peter"s day regarded this passage as messianic prophecy (cf. John 1:21 b, 25; Acts 7:40). Peter, by quoting this prophecy, affirmed that Jesus was the Messiah and urged his readers to accept Him or face destruction ( Acts 3:23). Destruction followed in A.D70. Belief in Moses should have led to belief in Jesus, and belief in Jesus would have made Peter"s hearers obedient to Moses. [source][source][source]
"The particular interest of this sermon lies in the way in which it gives further teaching about the person of Jesus, describing him as God"s servant, the Holy and Righteous One, the Author of life and the prophet like Moses. This indicates that a considerable amount of thinking about Jesus, based on study of the Old Testament, was taking place [2]." [3][source]