Can we move quantifiers to the left in predicate logic?

Question

Say I have part of a query in the form: ∃x_a(...)∧∃x_b(...)∧∃x_c(...), where a, b, and c are attributes and the ellipses can be anything (I'm looking for a general rule). Is this equivalent to saying ∃x_a,x_b,x_c(...∧...∧...) - i.e. compacting all the existential quantifiers into one and 'anding' their domains together?

For example, if I have the query:

would it be correct (albeit unwieldy) to write it as:

{ x_pid | ∃x_pname,x_color,x_sid,x_sname,x_address,x_cost,y_sid,y_sname,y_address,y_cost( PARTS(x_pid,x_pname,x_color) ∧ SUPPLIERS(x_sid,x_sname,x_address... (and then the rest of the query)