Theorem 19.9t 2189

Description: Closed form of 19.9 2190 and version of 19.3t 2185 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 22	. . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
2	1	19.9d 2187	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3		19.8a 2166	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
4	2, 3	impbid1 217	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 198  ∃wex 1823  Ⅎwnf 1827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163

This theorem depends on definitions:  df-bi 199  df-ex 1824  df-nf 1828

This theorem is referenced by:  19.9  2190  19.21t  2191  sbftv  2245  spimtOLD  2351  sbft  2455  vtoclegft  3482  bj-cbv3tb  33307  bj-spimtv  33314  bj-equsal1t  33392  bj-19.21t  33400