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Metamath Proof Explorer

Theorem pm3.24

Description: Law of noncontradiction. Theorem *3.24 of WhiteheadRussell p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993) (Proof shortened by Wolf Lammen, 24-Nov-2012)

		Ref	Expression
	Assertion	pm3.24	⊢ ¬ ( 𝜑 ∧ ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝜑 → 𝜑 )
2		iman	⊢ ( ( 𝜑 → 𝜑 ) ↔ ¬ ( 𝜑 ∧ ¬ 𝜑 ) )
3	1 2	mpbi	⊢ ¬ ( 𝜑 ∧ ¬ 𝜑 )