In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs — to prove an implication "A" → "B", assume "A" as an hypothesis and then proceed to derive "B" — in systems that do not have an explicit inference rule for this. Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style deduction systems because it permits one to write more comprehensible and usually much shorter proofs than would be possible without it. In certain other formal proof systems the same conveniency is provided by an explicit inference rule; for example natural deduction calls it implication introduction.