Theorem nf3

Description: Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021)

	Ref	Expression
Assertion	nf3	$$⊢Ⅎx\phi \leftrightarrow \left(\forall x\phi \vee \forall x\neg \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nf2	$$⊢Ⅎx\phi \leftrightarrow \left(\forall x\phi \vee \neg \exists x\phi \right)$$
2		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
3	2	orbi2i	$$⊢\left(\forall x\phi \vee \forall x\neg \phi \right)\leftrightarrow \left(\forall x\phi \vee \neg \exists x\phi \right)$$
4	1 3	bitr4i	$$⊢Ⅎx\phi \leftrightarrow \left(\forall x\phi \vee \forall x\neg \phi \right)$$