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Metamath Proof Explorer

Theorem impt

Description: Importation theorem pm3.1 (closed form of imp ) expressed with primitive connectives. (Contributed by NM, 25-Apr-1994) (Proof shortened by Wolf Lammen, 20-Jul-2013)

		Ref	Expression
	Assertion	impt	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ¬ ( 𝜑 → ¬ 𝜓 ) → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		simprim	⊢ ( ¬ ( 𝜑 → ¬ 𝜓 ) → 𝜓 )
2		simplim	⊢ ( ¬ ( 𝜑 → ¬ 𝜓 ) → 𝜑 )
3	2	imim1i	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ¬ ( 𝜑 → ¬ 𝜓 ) → ( 𝜓 → 𝜒 ) ) )
4	1 3	mpdi	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ¬ ( 𝜑 → ¬ 𝜓 ) → 𝜒 ) )