The impredicative $\forall$ operation in Girard's system F permits encodings of various inductive types, such as the natural numbers. However, these types fail to satisfy the relevant $\eta$-rules, and so, in dependent type theory, lack \emph{dependent} elimination rules. We work in Martin-L\"{o}f type theory with an impredicative universe. Using ideas from homotopy type theory, we refine the impredicative encodings so that the dependent elimination rules do hold. We construct a type of natural numbers as an initial algebra, which arises as a subtype of the System F encoding. We also encode some higher inductive types, such as the unit circle $S^1$.