Theorem nf6

Description: An alternate definition of df-nf . (Contributed by Mario Carneiro, 24-Sep-2016)

	Ref	Expression
Assertion	nf6	$$⊢Ⅎx\phi \leftrightarrow \forall x\left(\exists x\phi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-nf	$$⊢Ⅎx\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
2		nfe1	$$⊢Ⅎx\exists x\phi$$
3	2	19.21	$$⊢\forall x\left(\exists x\phi \to \phi \right)\leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
4	1 3	bitr4i	$$⊢Ⅎx\phi \leftrightarrow \forall x\left(\exists x\phi \to \phi \right)$$