Clarify if you realize this may refer to many false teachers and false messiahs ?
In Jewish context, "the" mssiah is considered the promised great King of Israel descended from David, who will lead Israel and the world into a time of peace and safety. Also he will bring all of the house of Israel back to the land of Israel.
Broadly, in Jewish context, messiah means anointed and applies to the kings and priests. "the" messiah is that one special anointed.
The verse John 5:43 is referring to "the:" messiah. Which Jesus is "the" messiah coming in the name of the Lord. But because of being lead astray by the religious leaders of the time, who Jesus was critical of, the Jews rejected Jesus as their King of Israel.
Before the tide was turned by the religious leaders, many of the Jews thought that Jesus was the promised messiah, King of Israel, as it says in John 12:13.
John 12:12-13
12 On the next day much people that were come to the feast, when they heard that Jesus was coming to Jerusalem,
13 Took branches of palm trees, and went forth to meet him, and cried, Hosanna: Blessed is the King of Israel that cometh in the name of the Lord.
So John 5:43 is in reference to another person that the Jews will accept, thinking he is their long awaited messiah, King of Israel. The Jews also believe to become the King of Israel, the person has to be anointed by a "known" prophet. Which they are also expecting Elijah to come to be a part of the messianic age start up.
In Revelation 13, the false prophet causes fire to come down from heaven - which Elijah also did in his battle with the prophets of Baal. For that reason, imo, the false prophet of Revelation 13, before he turns bad, like their perceived messiah the Antichrist, will present himself as Elijah, and will be the one who anoints the Antichrist person as the King of Israel. That person could be the Yanuka in Israel right now.
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Separately, Jesus did say that there would be false Christs (pretending to Him) and false prophets, in Matthew 24:24. Examples of such false Christs would be like José Luis de Jesús Miranda, Vissarion of Russia, John Miller of Australia.