Theorem 19.9tOLD 1857

Description: Obsolete proof of 19.9t 1779 as of 30-Dec-2017. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.)(New usage is discouraged.)

Assertion

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

Proof of Theorem 19.9tOLD

Step	Hyp	Ref	Expression
1		df-ex 1542	. . 3 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
2		id 19	. . . . . 6 ⊢ (Ⅎxφ → Ⅎxφ)
3	2	nfnd 1791	. . . . 5 ⊢ (Ⅎxφ → Ⅎx ¬ φ)
4	3	nfrd 1763	. . . 4 ⊢ (Ⅎxφ → (¬ φ → ∀x ¬ φ))
5	4	con1d 116	. . 3 ⊢ (Ⅎxφ → (¬ ∀x ¬ φ → φ))
6	1, 5	syl5bi 208	. 2 ⊢ (Ⅎxφ → (∃xφ → φ))
7		19.8a 1756	. 2 ⊢ (φ → ∃xφ)
8	6, 7	impbid1 194	1 ⊢ (Ⅎxφ → (∃xφ ↔ φ))

Syntax hints:  ¬ wn 3   → 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: (None)