Theorem pm2.18 102

Description: Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem pm2.18

Step	Hyp	Ref	Expression
1		pm2.21 100	. . . 4 ⊢ (¬ φ → (φ → ¬ (¬ φ → φ)))
2	1	a2i 12	. . 3 ⊢ ((¬ φ → φ) → (¬ φ → ¬ (¬ φ → φ)))
3	2	con4d 97	. 2 ⊢ ((¬ φ → φ) → ((¬ φ → φ) → φ))
4	3	pm2.43i 43	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:  pm2.18d  103  pm4.81  355  ax10  1944