[quote=@Catchphrase]
[@Legion-114] How is he an AI?
[/quote]

President Eden was an AI created as a monitoring system for Raven Rock, who eventually became self-aware. Only the Autumn family (technically a Scientist named Autumn and his son, Augustus), even knew that the President was actually an AI.

Depending on the choices made during Fallout 3, the player can convince the AI to destroy itself, but most people throughout the former US who even knew about Eden would assume that he was a real person. The reality, though is that Eden is/was an AI.

I assume that Legion is operating under the assumption that Eden was "killed" at Raven Rock, considering that our story here takes place either around or just following the events of F3 in 2277.

And, by that extension, Irons essentially has "rebooted" the president using this facility's technology, I assume.

[url]http://fallout.wikia.com/wiki/John_Henry_Eden[/url]