A formula , where denote either or and denotes either or (but no implication!), is equivalent to both and . (I put a period after a quantified variable when the scope of the quantifier extends as far to the right as possible.)Edit: The conclusion I've reached is that the order doesn't matter as long as we have predicates with only one variable. As for multiple variables, I'd assume the quantifiers have to be in the order their variables appear in the predicates. Can anyone confirm this?
Sometimes there are simpler prenex forms. is equivalent to and, dually, is equivalent to . This works because universal quantification is basically an infinite conjunction and the order of conjuncts does not matter.
In general, is equivalent to when and are either both universal or both existential. There is a huge difference between and . The important point here is the "interaction" of and .