Full Idea
Instead of demonstrations which are either axioms, or follow from axioms by rules, we can have one ever-growing sequence of formulas of the form 'Axioms |- ______', where the blank is filled by Axioms, then Lemmas, then Theorems, then Corollaries.
Gist of Idea
We can build one expanding sequence, instead of a chain of deductions
Source
John P. Burgess (Philosophical Logic [2009], 5.3)
Book Reference
Burgess,John P.: 'Philosophical Logic' [Princeton 2009], p.105