Theorem 19.9t

Description: Closed form of 19.9 and version of 19.3t with an existential quantifier. (Contributed by NM, 13-May-1993) (Revised by Mario Carneiro, 24-Sep-2016) (Proof shortened by Wolf Lammen, 14-Jul-2020)

	Ref	Expression
Assertion	19.9t	$$⊢Ⅎx\phi \to \left(\exists x\phi \leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢Ⅎx\phi \to Ⅎx\phi$$
2	1	19.9d	$$⊢Ⅎx\phi \to \left(\exists x\phi \to \phi \right)$$
3		19.8a	$$⊢\phi \to \exists x\phi$$
4	2 3	impbid1	$$⊢Ⅎx\phi \to \left(\exists x\phi \leftrightarrow \phi \right)$$