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Independence of premise
In proof theory and constructive mathematics, the principle of independence of premise states that if φ and ∃ ''x'' θ are sentences in a formal theory and is provable, then is provable. Here ''x'' cannot be a free variable of φ.
The principle is valid in classical logic. Its main application is in the study of intuitionistic logic, where the principle is not always valid.
== In classical logic ==
The principle of independence of premise is valid in classical logic because of the law of the excluded middle. Assume that is provable. Then, if φ holds, there is an ''x'' satisfying φ → θ but if φ does not hold then ''any'' ''x'' satisfies φ → θ. In either case, there is some ''x'' such that φ→θ. Thus is provable.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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