What is the symmetry between the definitions of the bounded universal/existential quantifiers? What is the symmetry between the definitions of the bounded universal/existential quantifiers?
$\forall x \in A, B(x)$ means $\forall x (x \in A \rightarrow B(x))$
$\exists x \in A, B(x)$ means $\exists x (x \in A \land B(x))$
These make intuitive sense, but I would expect there to be some kind of symmetry between how the definitions of the bounded quantifiers work, and I can't see one. $A \rightarrow B$ means $\lnot A \lor B$ which doesn't seem to have a direct relationship with $A \land B$. What am I missing? 

As Asaf Karagila says, those two connectives, $\forall x \in A, \ldots$ and $\exists x \in A, \ldots$ are dual of each other.
Just like the conjunction / disjunction pair or the universal / existential pair, we have that $\lnot (\forall x \in A, \phi(x)) \Leftrightarrow \exists x \in A, \lnot \phi(x)$.
Also, if $A$ is a singleton $\{a\}$, those two connective are logically equivalent, and $\forall x \in A, \phi(x) \Leftrightarrow \exists x \in A, \phi(x) \Leftrightarrow \phi(a)$
