# Deduction theorem

In mathematical logic, the deduction theorem states that if a formula F is deducible from E then the implication E → F is demonstrable (i.e. it is "deducible" from the empty set). In symbols, if [itex] E \vdash F [itex], then [itex] \vdash E \rightarrow F. [itex]

The deduction theorem may be generalized to a countable sequence of assumption formulas such that from

[itex] E_1, E_2, ... , E_{n-1}, E_n \vdash F [itex], infer [itex] E_1, E_2, ... , E_{n-1} \vdash E_n \rightarrow F [itex], and so on until

[itex] \vdash E_1\rightarrow(...(E_{n-1} \rightarrow (E_n \rightarrow F))...) [itex].

The deduction theorem is a meta-theorem: it is used to deduce proofs in a given theory though it is not a theorem of the theory itself.

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