Theorem 19.9t 2246

Description: Closed form of 19.9 2247 and version of 19.3t 2243 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.)

Assertion

Ref	Expression
19.9t	⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

Proof of Theorem 19.9t

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
2	1	19.9d 2245	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3		19.8a 2223	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
4	2, 3	impbid1 228	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-ex 1807  df-nf 1811

This theorem is referenced by:  19.9  2247  19.21t  2248  sbft  2311  bj-cbv3tb  37310  bj-spimtv  37317  bj-equsal1t  37345  bj-19.21t0  37353  19.9dev  42875