In more detail, the propositional logic deduction theorem states that if a formula formula_1 is deducible from a set of assumptions formula_2 then the implication formula_3 is deducible from formula_4; in symbols, formula_5 implies formula_6. In the special case where formula_4 is the empty set, the deduction theorem claim can be more compactly written as: formula_8 implies formula_9. The deduction theorem for predicate logic is similar, but comes with some extra constraints (that would for example be satisfied if formula_10 is a closed formula). In general a deduction theorem needs to take into account all logical details of the theory under consideration, so each logical system technically needs its own deduction theorem, although the differences are usually minor.