Theorem pm2.18 129

Description: Clavius 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 pm2.01 190. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 17-Nov-2023.)

Assertion

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

Proof of Theorem pm2.18

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → 𝜑))
2	1	pm2.18d 128	1 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  pm2.18i  130  notnotr  131  pm4.81  399  sumdmdlem2  32818  axc11n11r  37341