Theorem 19.9d

Description: A deduction version of one direction of 19.9 . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 24-Sep-2016) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by Wolf Lammen, 8-Jul-2022)

	Ref	Expression
Hypothesis	19.9d.1	$$⊢\psi \to Ⅎx\phi$$
Assertion	19.9d	$$⊢\psi \to \left(\exists x\phi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.9d.1	$$⊢\psi \to Ⅎx\phi$$
2	1	nfrd	$$⊢\psi \to \left(\exists x\phi \to \forall x\phi \right)$$
3		sp	$$⊢\forall x\phi \to \phi$$
4	2 3	syl6	$$⊢\psi \to \left(\exists x\phi \to \phi \right)$$