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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 φ → ∃ x θ is provable, then ∃ x (φ → θ) 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.

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  • Independence of premise (en)
  • Indepêndencia de premissas (pt)
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  • 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 φ → ∃ x θ is provable, then ∃ x (φ → θ) 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. (en)
  • Na teoria da prova e matemática construtiva, o princípio da independência de premissas afirma que, se φ e ∃ x θ são sentenças em uma teoria formal e φ → ∃ x θ é demonstrável, então ∃ x (φ → θ) é demonstrável. Neste caso x não pode ser uma variável livre de φ. A sua principal aplicação é no estudo da lógica intuicionística, onde o princípio nem sempre é válido. (pt)
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  • 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 φ → ∃ x θ is provable, then ∃ x (φ → θ) 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. (en)
  • Na teoria da prova e matemática construtiva, o princípio da independência de premissas afirma que, se φ e ∃ x θ são sentenças em uma teoria formal e φ → ∃ x θ é demonstrável, então ∃ x (φ → θ) é demonstrável. Neste caso x não pode ser uma variável livre de φ. A sua principal aplicação é no estudo da lógica intuicionística, onde o princípio nem sempre é válido. (pt)
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