Can ∃ be defined using ∀?

Question

On this page explaining godel numbers (https://plato.stanford.edu/entries/goedel-incompleteness/sup1.html) it says "for simplicity, let us assume that ¬,→ and ∀ are the only primitive logical symbols, and that ∧,∨,↔ and ∃ are defined with the help of them". But I cant find how symbols like the existential quantifier can be defined from the symbols given?

Answer 1

$\exists$ is equivalent to $\neg \forall \neg$ (there exists a true instance if it is not the case that the statement always fails).

Equivalently, one can define $\forall$ as $\neg \exists \neg$.

Also, $a \vee b$ is equivalent to $\neg a \to b$. And, $a \wedge b$ is equivalent to $\neg(a \to \neg b)$. Finally, $a \leftrightarrow b$ is equivalent to $(a \to b) \wedge (b \to a)$ which merely abbreviates $\neg((a \to b) \to \neg(b \to a))$.