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Metamath Proof Explorer

Theorem anor

Description: Conjunction in terms of disjunction (De Morgan's law). Theorem *4.5 of WhiteheadRussell p. 120. (Contributed by NM, 3-Jan-1993) (Proof shortened by Wolf Lammen, 3-Nov-2012)

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

Proof

Step	Hyp	Ref	Expression
1		notnotb	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ¬ ( 𝜑 ∧ 𝜓 ) )
2		ianor	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )
3	1 2	xchbinx	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ( ¬ 𝜑 ∨ ¬ 𝜓 ) )