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Metamath Proof Explorer

Theorem ianor

Description: Negated conjunction in terms of disjunction (De Morgan's law). Theorem *4.51 of WhiteheadRussell p. 120. (Contributed by NM, 14-May-1993) (Proof shortened by Andrew Salmon, 13-May-2011)

		Ref	Expression
	Assertion	ianor	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		imnan	⊢ ( ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
2		pm4.62	⊢ ( ( 𝜑 → ¬ 𝜓 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )
3	1 2	bitr3i	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )