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A derived inference rule abbreviates a logical derivation from the primitive inference rules. Its use retains the assumptions and side conditions required by that logical derivation; proving the abbreviation once allows subsequent proofs to use the rule without expanding every intermediate line.

An inference rule licenses a conclusion from specified premises, possibly with side conditions. A derived inference rule is justified by a logical derivation using the primitive inference rules; using it subsequently abbreviates that logical derivation.

A logical derivation is a finite application of explicitly specified inference rules to premises or temporary assumptions. In natural deduction, which assumptions remain open is part of the logical derivation: a conclusion reached inside a temporary subproof cannot simply be used outside it.

A temporary assumption is introduced for a hypothetical subproof and retained only within that scope. A suitable inference rule can discharge it while forming a conclusion valid under the surrounding assumptions. Its scope must be recorded even when the assumption is unused inside the subproof.