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I would say that "Anyone can cook" means that it is possible for any person to learn to cook, but that "Everybody can cook" means that all people know how to cook.
In a more restricted sense, if this is said in a kitchen or a classroom, then any person there is allowed to cook OR everybody there is allowed to cook.
So am I correct to assume that to lower-intermediate EASL learners in a classroom there won't be any difference at all. (I'm sorry I'm a bit slow tonight)
anyone 
(I'm sorry I'm a bit slow tonight)
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