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	$$⊢Ⅎx\phi \leftrightarrow Ⅎx\neg \phi$$

Proof

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