From wikipedia quantification has meaning:

In logic, quantification is the binding of a variable ranging over a domain of discourse

Is there any formal "definition" of universal quantifier for example using definition of domain of discourse?

I mean a formula build without universal quantifier, and existential one which has the same meaning if referenced to defined domain of discourse?

For example: Suppose we use domain of discourse (DoD) given by sentence $ U = \{ x|\phi(x) \}$ for some $\phi(x)$. Then naively we may wrote:

($\forall (x \in U) \Phi(x) ) \equiv ( \{ x|\phi(x) \} => \Phi(x) )$

In words: to say that some property follows for every x in DoD is the same as to say that if x is chosen from DoD then has this property.

We may try also the folowing one: ($\forall (x \in U) \Phi(x) ) \equiv (( \{ x|\phi(x) \} => \Phi(x) ) => (\phi(x) <=> \Phi(x) ))$

In words: to say that some property follows for every x in DoD is the same as to say that $\phi$ and $\Phi$ are evenly spanned.

Do You know any reference for such matter?