I had not thought about the possibility of the False Prophet in Rev. 13 & Rev. 19:20 pretending to be Elijah the Prophet until I read David Jeremiah's article about it at Will Elijah Return During the Tribulation? - Living in the Age of Signs - DavidJeremiah.org  He makes a point about Elijah bringing fire down from heaven (2 Kings 1:10 & 12) and the False Prophet also does so according to Rev. 13:13.  (Please read the entire article at the above link.)

The Jewish people also expect Elijah the prophet before the coming of the Messiah because Malachi 4:5 says, "Behold, I will send you Elijah the prophet before the coming of the great and dreadful day of the LORD: And he shall turn the heart of the fathers to the children, and the heart of the children to their fathers, lest I come and smite the earth with a curse."  

David Jeremiah's article stated at the above link that "Revelation 11 describes two witnesses who will preach to the world after the Rapture. After the Church is taken away, God will raise up 144,000 new believers and appoint the two witnesses to preach the Gospel and God's judgment to everyone who is left to hear it. Through these witnesses, God will continue to provide opportunities for people to place their faith in Him right up until the end of the world."

However, if the False Prophet pretends to be Elijah the Prophet, he could very well with his signs and lying wonders deceive the people into believing that the Antichrist is the Messiah who is the one in Isaiah 53 who was wounded for their transgressions who is cut off out of the land of the living when they see the Antichrist wounded unto death by a sword and then he comes to life again. (Rev. 13:3,12-14).