Is predicate logic an extension of propositional logic? As I read in this thread, the difference between propositional and predicate logic is that in predicate logic you can use things like quantifiers, predicates and functions whereas in propositional logic you cannot. However, you can use things like letters, $\wedge$, $\vee$, $\neg$ in _both_ propositional and predicate logic. Does this mean that everything we can use in propositional logic we can use in predicate logic? In other words, propositional logic $\subset$ predicate logic?

Yes, that is correct!

Propositional logic studies what is logically true or implied on the basis of truth-functional operators ($\land$, $\lor$, $\
eg$, etc)

Predicate logic studies what is logically true or implied on the basis of truth-functional operators ($\land$, $\lor$, $\
eg$, etc) _and_ predicates, _and_ individual constants, _and_ quantifiers (using variables).

If you throw in identity as well, you get first-order logic (some people make a distinction between predicate logic and first-order logic)