deduction theorem

deduction theorem (plural deduction theorems)

1. (logic) A procedure for "discharging" assumptions from an inference, causing them to become antecedents of the conclusion; or vice versa. Symbolically, the conversion of an inference of the form ${\displaystyle P,A\vdash C}$ to an inference of the form ${\displaystyle P\vdash A\rightarrow C}$ or vice versa, where ${\displaystyle \vdash }$ is the turnstile symbol. The validity of the procedure is a metatheorem of the given logical theory.

   The deduction theorem reveals the relationship between logical entailment and material implication: it allows to one to "pack" or "record" an inference into a tautology, and conversely, to "unpack" or "play back" a tautology as an inference process.

• conditional proof