A number of authors speak of "de Morgan's laws for quantifiers," and I think this is a wise choice of terminology. A universal (respectively, existential) quantifier behaves just like a conjunction (respectively, disjunction) over all the objects in the universe, so, aesthetically and pedagogically, I think it's much more elegant to speak of ¬∃x(P(x)) <---> ∀x(¬P(x)) and ¬∀x(P(x)) <---> ∃x(¬P(x)) as generalized de Morgan's laws, rather than to reserve the term "de Morgan's laws" for ¬(A ∧ B) <---> (¬A ∨ ¬B) and ¬(A ∨ B) <---> (¬A ∧ ¬B) and have a separate term like "quantifier negation laws" for the tautologies involving quantifiers. Because, you know, it's the same idea in slightly different guises. Some authors may prefer different terminology, but I stand by my comment.