Theorem nf4

Description: Alternate definition of nonfreeness. This definition uses only primitive symbols ( -> , -. , A. ). (Contributed by BJ, 16-Sep-2021)

	Ref	Expression
Assertion	nf4	$$⊢Ⅎx\phi \leftrightarrow \left(\neg \forall x\phi \to \forall x\neg \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nf3	$$⊢Ⅎx\phi \leftrightarrow \left(\forall x\phi \vee \forall x\neg \phi \right)$$
2		df-or	$$⊢\left(\forall x\phi \vee \forall x\neg \phi \right)\leftrightarrow \left(\neg \forall x\phi \to \forall x\neg \phi \right)$$
3	1 2	bitri	$$⊢Ⅎx\phi \leftrightarrow \left(\neg \forall x\phi \to \forall x\neg \phi \right)$$