Theorem 19.9t 1573

Description: A closed version of 19.9 1575. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)

Assertion

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

Proof of Theorem 19.9t

Step	Hyp	Ref	Expression
1		df-nf 1390	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		19.9ht 1572	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
3	1, 2	sylbi 119	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
4		19.8a 1522	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
5	3, 4	impbid1 140	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282  Ⅎwnf 1389  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  19.9d  1591  19.23t  1607  spimt  1664  exdistrfor  1721  sbequi  1760  sbft  1769  vtoclegft  2670  copsexg  3999