Theorem 19.9t 2021

Description: A closed version of 19.9 2022. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (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 2020	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3		19.8a 1988	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
4	2, 3	impbid1 213	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 194  ∃wex 1694  Ⅎwnf 1698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983

This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1699

This theorem is referenced by:  19.9  2022  19.21t  2035  spimt  2144  sbft  2271  vtoclegft  3157  bj-cbv3tb  31736  bj-spimtv  31743  bj-sbftv  31793  bj-equsal1t  31842  bj-19.21t  31850  19.9alt  33164