Theorem pm2.18 125

Description: Clavius's law, or "consequentia mirabilis" ("admirable consequence"). If a formula is implied by its negation, then it is true. Can be used in proofs by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. See also the weak Clavius law pm2.01 181. (Contributed by NM, 29-Dec-1992.)

Assertion

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

Proof of Theorem pm2.18

Step	Hyp	Ref	Expression
1		pm2.21 121	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → ¬ (¬ 𝜑 → 𝜑)))
2	1	a2i 14	. . 3 ⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑)))
3	2	con4d 115	. 2 ⊢ ((¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑))
4	3	pm2.43i 52	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  126  pm2.18d  127  notnotr  128  pm4.81  384  sumdmdlem2  29850  axc11n11r  33262