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	⊢ ( Ⅎ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( Ⅎ 𝑥 𝜑 → Ⅎ 𝑥 𝜑 )
2	1	19.9d	⊢ ( Ⅎ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )
3		19.8a	⊢ ( 𝜑 → ∃ 𝑥 𝜑 )
4	2 3	impbid1	⊢ ( Ⅎ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 ↔ 𝜑 ) )