Theorem nf5

Description: Alternate definition of df-nf . (Contributed by Mario Carneiro, 11-Aug-2016) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021)

	Ref	Expression
Assertion	nf5	$$⊢Ⅎx\phi \leftrightarrow \forall x\left(\phi \to \forall x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-nf	$$⊢Ⅎx\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
2		nfa1	$$⊢Ⅎx\forall x\phi$$
3	2	19.23	$$⊢\forall x\left(\phi \to \forall x\phi \right)\leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
4	1 3	bitr4i	$$⊢Ⅎx\phi \leftrightarrow \forall x\left(\phi \to \forall x\phi \right)$$