After the MRDP theorem, it was widely believed that most arithmetically significant rings andfields have first-order definable sets too complicated to understand fully, even at the lowest non- trivial complexity. Pop formulated a more precise problem called Elementary Equivalence Versus Isomorphism, which asked whether these rings and fields can detect their own isomorphism class among all other arithmetically significant rings and fields using only first-order properties. An extensive body of work had resolved many cases of this problem. In this thesis, we reinforce this intuition by confirming another case, showing that for a function field of transcendence degree 2 over a local field with residue characteristic not 2, the strong Elementary Equivalence Versus Isomorphism assertion is true. If K|k is a function field over a local field k of transcendence degree 2, then there is a first-order formula ϕ such that for any function field over local field K′|k′, ϕ is true on K′ if and only if K ∼= K′.