# Is existential quantifier redundant

I was reading Tuple Relational Calculus from a textbook. I came across this query:

For every project located in 'Stafford', list the project number, the controlling department number, and the department manager's last name, birth date and address.

The solution query is:

{p.Pnumber,p.Dnum,m.Lname,m.Bdate,m.Address |
PROJECT(p)}  AND
EMPLOYEE(m)  AND
p.Plocation = 'Stafford' AND
((∃d)(DEPARTMENT(d) AND
p.Dnum = d.Dnumber AND
d.Mgr_ssn = m.Ssn
)
)
}


I dont understand what difference existential quantifier will make if I can write the same query as follows without existential quantifier (and of course if it produces same result and hence correct, which also I dont know):

{p.Pnumber,p.Dnum,m.Lname,m.Bdate,m.Address |
PROJECT(p)}  AND
EMPLOYEE(m)  AND
DEPARTMENT(d)
p.Plocation = 'Stafford' AND
p.Dnum = d.Dnumber AND
d.Mgr_ssn = m.Ssn
}


Is it like: the solution query provided uses existential quantifier just because we dont actually project DEPARTMENT relation's attributes in the final output?