Relational Algebra: existential quantifiers
Let me concentrate, as you asked, on section $(\exists d)(\exists t)(\cdots)$.
In case when one part of conjuction is &quot;not depend&quot; on letter under existence quantifier, holds
$$(\exists d)(A \text{ and } B)\Leftrightarrow  (\exists d)(A) \text{ and } B$$
Where we assume $B$ does not containt letter $d$.
$$$$
In your case, for simplicity, let make following designations:
$$\begin{array}{l}
A(d) = \text{ DEPARTMENT(d) }\\
B(t) = \text{ DEPENDANT(t) } \\
C(d) = (\text{e.Ssn = d.Mgrssn})\\
E(t) = (\text{t.Essn = e.Ssn})
\end{array}$$
and assume, that $A,C$ doesn't depend on $t$ and $B,E$ doesn't depend on $d$. Then we have
$$(\exists d)(\exists t)\big(A(d) \text{ and } B(t)\text{ and } C(d) \text{ and } E(t)\big) \Leftrightarrow \\
\Leftrightarrow (\exists d)\big(A(d) \text{ and } C(d)\big) \text{ and } (\exists t) \big(B(t)\text{ and } E(t)\big)$$
So, you are right, they are equivalent in reviewed case.
Generally we cannot distribute existence quantifier  with respect to $\text{AND}$(conjunction) operation as shows example: take $A=$&quot;$d$ is even&quot; and $B=$&quot;$d$ is odd&quot;. Then, of course, sentence &quot;$(\exists d)(A) \text{ and } (\exists d)(B)$&quot; is true, but sentence &quot;$(\exists d)(A \text{ and } B)$&quot; is false.