Theorem pm2.18 121

Description: Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. See also pm2.01 179. (Contributed by NM, 29-Dec-1992.)

Assertion

Ref	Expression
pm2.18	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Proof of Theorem pm2.18

Step	Hyp	Ref	Expression
1		pm2.21 119	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → ¬ (¬ 𝜑 → 𝜑)))
2	1	a2i 14	. . 3 ⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑)))
3	2	con4d 113	. 2 ⊢ ((¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑))
4	3	pm2.43i 50	1 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

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:  pm2.18i  122  pm2.18d  123  pm4.81  380  sumdmdlem2  28662  pm4.81ALT  31716  axc11n11r  31860