Consider the quantifier "for every" , it simply means variable '$x$' has to be true for all values of '$x$' upon a propositional function $P(x)$. So we could 'AND' all the values from domain of 'x' and find whether its true.
For example: $\forall x[P(x)]$ is equivalent to : $P(x_1) \land P(x_2) \land \dots \land P(x_n)$
Similar case for "there exists", atleast one of them has to be true. So OR could be used
$\exists x[P(x)]$ is equivalent to : $P(x_1) \lor P(x_2) \lor \dots \lor P(x_n)$
What I am asking is when it comes to proposition involving more than one quantifiers. How to express it in above form ?
For example, consider a proposition involving 2 variables $P(x,y)$. Consider $\exists x \forall x[P(x,y)]$. How can I write it using 'AND' and 'OR'