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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	$⊢ (φ \to (ψ \to χ)) \to (\neg (φ \to \neg ψ) \to χ)$

Proof

Step	Hyp	Ref	Expression
1		simprim	$⊢ \neg (φ \to \neg ψ) \to ψ$
2		simplim	$⊢ \neg (φ \to \neg ψ) \to φ$
3	2	imim1i	$⊢ (φ \to (ψ \to χ)) \to (\neg (φ \to \neg ψ) \to (ψ \to χ))$
4	1 3	mpdi	$⊢ (φ \to (ψ \to χ)) \to (\neg (φ \to \neg ψ) \to χ)$