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Metamath Proof Explorer

Theorem pm2.01

Description: Weak Clavius law. If a formula implies its negation, then it is false. A form of "reductio ad absurdum", which can be used in proofs by contradiction. Theorem *2.01 of WhiteheadRussell p. 100. Provable in minimal calculus, contrary to the Clavius law pm2.18 . (Contributed by NM, 18-Aug-1993) (Proof shortened by Mel L. O'Cat, 21-Nov-2008) (Proof shortened by Wolf Lammen, 31-Oct-2012)

		Ref	Expression
	Assertion	pm2.01	⊢ ( ( 𝜑 → ¬ 𝜑 ) → ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ¬ 𝜑 → ¬ 𝜑 )
2	1 1	ja	⊢ ( ( 𝜑 → ¬ 𝜑 ) → ¬ 𝜑 )