Theorem 19.9t 1621

Description: A closed version of 19.9 1623. (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 1437	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		19.9ht 1620	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
3	1, 2	sylbi 120	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))
4		19.8a 1569	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
5	3, 4	impbid1 141	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  19.9d  1639  19.23t  1655  spimt  1714  exdistrfor  1772  sbequi  1811  sbft  1820  vtoclegft  2753  copsexg  4161