Deduction-theorem meaning

(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 to an inference of the form or vice versa, where 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.