Theorem nf5r

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by Wolf Lammen, 23-Nov-2023)

	Ref	Expression
Assertion	nf5r	$$⊢Ⅎx\phi \to \left(\phi \to \forall x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.8a	$$⊢\phi \to \exists x\phi$$
2		id	$$⊢Ⅎx\phi \to Ⅎx\phi$$
3	2	nfrd	$$⊢Ⅎx\phi \to \left(\exists x\phi \to \forall x\phi \right)$$
4	1 3	syl5	$$⊢Ⅎx\phi \to \left(\phi \to \forall x\phi \right)$$