Theorem 19.9t 2109

Description: A closed version of 19.9 2110. (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 2108	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
3		19.8a 2090	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
4	2, 3	impbid1 215	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196  ∃wex 1744  Ⅎwnf 1748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750

This theorem is referenced by:  19.9  2110  19.21t  2111  19.21tOLDOLD  2112  spimt  2289  sbft  2407  vtoclegft  3311  bj-cbv3tb  32836  bj-spimtv  32843  bj-sbftv  32888  bj-equsal1t  32934  bj-19.21t  32942  19.9alt  34570