Does ~∃(x) A(x) equal to ∀(x) ~A(x)?
The question is whether $\displaystyle {\sim}\exists (x)A\,(x)$ is equal to $\displaystyle \forall(x)\,{\sim} A(x)$. The short answer is yes. More precisely, it depends on the definition of "equal". I would say that, since these are two different formulas, they are not equal but equivalent, meaning that they are true in the same interpretations, or that they are derivable from each other.