Full Idea
Complete proof procedures are available not only for quantification theory, but for quantification theory and identity together. Gödel showed that the theory is still complete if we add self-identity and the indiscernability of identicals.
Gist of Idea
Quantification theory can still be proved complete if we add identity
Source
Willard Quine (Philosophy of Logic [1970], Ch.5)
Book Reference
Quine,Willard: 'Philosophy of Logic' [Prentice-Hall 1970], p.62
A Reaction
Hence one talks of first-order logic 'with identity', even though, as Quine observes, it is unclear whether identity is actually a logical or a mathematical notion.