Theorem nf5-1

Description: One direction of nf5 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021)

	Ref	Expression
Assertion	nf5-1	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to Ⅎx\phi$$

Proof

Step	Hyp	Ref	Expression
1		exim	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\exists x\phi \to \exists x\forall x\phi \right)$$
2		hbe1a	$$⊢\exists x\forall x\phi \to \forall x\phi$$
3	1 2	syl6	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\exists x\phi \to \forall x\phi \right)$$
4	3	nfd	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to Ⅎx\phi$$