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Metamath Proof Explorer

Theorem nfnbi

Description: A variable is nonfree in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019) (Proof shortened by Wolf Lammen, 6-Oct-2024)

		Ref	Expression
	Assertion	nfnbi	$⊢ Ⅎ x φ \leftrightarrow Ⅎ x \neg φ$

Proof

Step	Hyp	Ref	Expression
1		exnal	$⊢ \exists x \neg φ \leftrightarrow \neg \forall x φ$
2	1	imbi1i	$⊢ (\exists x \neg φ \to \forall x \neg φ) \leftrightarrow (\neg \forall x φ \to \forall x \neg φ)$
3		df-nf	$⊢ Ⅎ x \neg φ \leftrightarrow (\exists x \neg φ \to \forall x \neg φ)$
4		nf4	$⊢ Ⅎ x φ \leftrightarrow (\neg \forall x φ \to \forall x \neg φ)$
5	2 3 4	3bitr4ri	$⊢ Ⅎ x φ \leftrightarrow Ⅎ x \neg φ$