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Metamath Proof Explorer

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	⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )

Proof

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