Full Idea
Frege's formal definition of derivability is perhaps the first investigation in general proof theory.
Gist of Idea
Proof theory began with Frege's definition of derivability
Source
report of Gottlob Frege (Begriffsschrift [1879]) by Dag Prawitz - Gentzen's Analysis of First-Order Proofs 2 n2
Book Reference
'A Philosophical Companion to First-Order Logic', ed/tr. Hughes,R.I.G. [Hackett 1993], p.207
A Reaction
In 'On General Proof Theory §1' Prawitz says "proof theory originated with Hilbert" in 1900. Presumably Frege offered a theory, and then Hilbert saw it as a general project.