If we have the proposition
\begin{align} &\text{Bob is a Babylonian}\\ &\text{Bob is a Human}\\ &\text{Therefore, some Humans are Babylonians} \end{align}
which translates to
\begin{align} &B(p)\\ &H(p)\\ &\text{Therefore, } \exists x (B(x) \land H(x)) \end{align}
how would one prove such a proposition? Would it look something like
EDIT
\begin{align} &B(p) ... (\text{premise a})\\ &H(p) ... (\text{premise b})\\ &\text{Therefore, } B(p) \land H(p) ... (1) \text{ by premise a and b}) \\ &\text{Therefore, }\exists x (B(x) \land H(x)) ... (2) \text{ by (1)} \end{align}
It is predicate logic; thus, we need the rules for managing quantifiers.
The details depend on the proof system you are using... but basically we have to use the principle of Existential generalization.
From $B(p)$ and $H(p)$ we get $B(p) \land H(p)$ by propositional rules (specifically : Conjunction), that means that "object $p$ is both $B$ and $H$".
Then we can conclude that "there is something that is both $B$ and $H$".