Co-NP reduced in polynomial-time

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Dragonfall
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Every language in co-NP can be reduced in polynomial-time to an instance of [itex]\forall x P(x)[/itex] for some Boolean formula with a single bound variable and no free variables, right?
 
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A definition of a language being in co-NP would be that such an instance exists and the predicate is polynomial-time computable. I'm not sure if this is what you are getting at.