Theorem nf2

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

	Ref	Expression
Assertion	nf2	$$⊢Ⅎx\phi \leftrightarrow \left(\forall x\phi \vee \neg \exists x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-nf	$$⊢Ⅎx\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
2		imor	$$⊢\left(\exists x\phi \to \forall x\phi \right)\leftrightarrow \left(\neg \exists x\phi \vee \forall x\phi \right)$$
3		orcom	$$⊢\left(\neg \exists x\phi \vee \forall x\phi \right)\leftrightarrow \left(\forall x\phi \vee \neg \exists x\phi \right)$$
4	1 2 3	3bitri	$$⊢Ⅎx\phi \leftrightarrow \left(\forall x\phi \vee \neg \exists x\phi \right)$$