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At the end of RotS, Obi-Wan believes that he is leaving Anakin for dead on Mustafar, disappearing to Tatooine. While he knows that Anakin is not entirely dead at the end of their battle, he leaves Anakin's death to the Force. However, when Obi-Wan faces Vader in ANH, it is established that he knows exactly who he is facing. In RotJ, he explicitly tells Luke that he lied about Vader killing Anakin.

So, when did Obi-Wan discover that Anakin had become Vader?

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Obi-Wan and Yoda both witnessed a holo-recording of the newly self-crowned Emperor Palpatine addressing Anakin as "Lord Vader" right after they finished reprogramming the recall signal in "Episode 3: Revenge of the Sith". See the video below (at approx the 55 second mark).

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