Theorem 19.9t 1779

Description: A closed version of 19.9 1783. (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 1545	. . 3 ⊢ (Ⅎxφ ↔ ∀x(φ → ∀xφ))
2		19.9ht 1778	. . 3 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))
3	1, 2	sylbi 187	. 2 ⊢ (Ⅎxφ → (∃xφ → φ))
4		19.8a 1756	. 2 ⊢ (φ → ∃xφ)
5	3, 4	impbid1 194	1 ⊢ (Ⅎxφ → (∃xφ ↔ φ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.9h  1780  19.9d  1782  19.9OLD  1784  19.21t  1795  19.23t  1800  19.23tOLD  1819  vtoclegft  2926