Theorem expt 177

Description: Exportation theorem pm3.3 449 (closed form of ex 413) expressed with primitive connectives. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Garrett Katz, 25-May-2026.)

Assertion

Ref	Expression
expt	⊢ ((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))

Proof of Theorem expt

Step	Hyp	Ref	Expression
1		pm3.2im 160	. 2 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
2		id 22	. 2 ⊢ ((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (¬ (𝜑 → ¬ 𝜓) → 𝜒))
3	1, 2	syl9r 78	1 ⊢ ((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)