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Metamath Proof Explorer

Theorem nbn

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993) (Proof shortened by Wolf Lammen, 3-Oct-2013)

		Ref	Expression
	Hypothesis	nbn.1	$⊢ \neg φ$
	Assertion	nbn	$⊢ \neg ψ \leftrightarrow (ψ \leftrightarrow φ)$

Proof

Step	Hyp	Ref	Expression
1		nbn.1	$⊢ \neg φ$
2		bibif	$⊢ \neg φ \to ((ψ \leftrightarrow φ) \leftrightarrow \neg ψ)$
3	1 2	ax-mp	$⊢ (ψ \leftrightarrow φ) \leftrightarrow \neg ψ$
4	3	bicomi	$⊢ \neg ψ \leftrightarrow (ψ \leftrightarrow φ)$