why this formulas are equivalent? $\exists x \phi(x) \equiv \neg \forall x \neg \phi(x)$

Question

I can't understand why this formulas are equivalent, if exists a x have the property why is equivalent to the opposite?

Answer 1

Intuitively:

$\neg\forall x \neg\phi(x)$ means that not for all elements $x$ in a certain domain, we know that a statement $\phi(x)$ does not hold. If not for all elements the statement does not hold, there there must be an element for which the statement holds.

Answer 2

the equivalence is misstated ." there exists x Phi(x)" is equivalent to

"NOT for every x NOT Phi(x) . Someone forgot the "NOT" Stuart M.N. It looks now like all the forgotten " NOT"'S have been corrected . (including mine ).

Answer 3

Try to analise it in human language: LHS means "there exists $x$ such that $\phi(x)$", RHS is "it is false that for all $x$, $\phi(x)$ does not hold". To see it more formally, it is application of De Morgan's laws to predicate logic.