Theorem nfnbi

Description: A variable is non-free in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019)

	Ref	Expression
Assertion	nfnbi	$$⊢Ⅎx\phi \leftrightarrow Ⅎx\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		orcom	$$⊢\left(\forall x\phi \vee \forall x\neg \phi \right)\leftrightarrow \left(\forall x\neg \phi \vee \forall x\phi \right)$$
2		nf3	$$⊢Ⅎx\phi \leftrightarrow \left(\forall x\phi \vee \forall x\neg \phi \right)$$
3		nf3	$$⊢Ⅎx\neg \phi \leftrightarrow \left(\forall x\neg \phi \vee \forall x\neg \neg \phi \right)$$
4		notnotb	$$⊢\phi \leftrightarrow \neg \neg \phi$$
5	4	albii	$$⊢\forall x\phi \leftrightarrow \forall x\neg \neg \phi$$
6	5	orbi2i	$$⊢\left(\forall x\neg \phi \vee \forall x\phi \right)\leftrightarrow \left(\forall x\neg \phi \vee \forall x\neg \neg \phi \right)$$
7	3 6	bitr4i	$$⊢Ⅎx\neg \phi \leftrightarrow \left(\forall x\neg \phi \vee \forall x\phi \right)$$
8	1 2 7	3bitr4i	$$⊢Ⅎx\phi \leftrightarrow Ⅎx\neg \phi$$