Theorem pm2.18

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 . (Contributed by NM, 29-Dec-1992) (Proof shortened by Wolf Lammen, 17-Nov-2023)

	Ref	Expression
Assertion	pm2.18	$$⊢\left(\neg \phi \to \phi \right)\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\left(\neg \phi \to \phi \right)\to \left(\neg \phi \to \phi \right)$$
2	1	pm2.18d	$$⊢\left(\neg \phi \to \phi \right)\to \phi$$