Isn't reincarnation affirmed by the Bible?

Malachi 4:5: >I will send you the prophet Elijah. He will come before the day of the Lord arrives. It will be a great and terrifying day Jesus says in Matthew 11:14 >"and if you are willing to believe their message, John is Elijah, whose coming was predicted" Jesus says in Mathew 17:12 >But I tell you, Elijah has already come, and they did not recognize him, but have done to him everything they wished. In the same way the Son of Man is going to suffer at their hands.” It's pretty clear from the above verses that John was Elijah reincarnated. Wouldn't the above verses imply that reincarnation is true?