CAIRO—The most bitter rivalry emerging in Egypt's presidential campaign pits two candidates with competing visions of Islam and a long-running personal feud.

The race between the Muslim Brotherhood's bid to win the presidency and Abdel Moneim Aboul Fotouh, who split from the group last summer, could help define the contest over the next five weeks until the May 23 election day. Mr. Aboul Fotouh's more liberal vision of Islam has long been at odds with the Brotherhood's conservative old guard.

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