Theorem 19.9ht 1778

Description: A closed version of 19.9 1783. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.9ht	⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))

Proof of Theorem 19.9ht

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
2		hbnt 1775	. . 3 ⊢ (∀x(φ → ∀xφ) → (¬ φ → ∀x ¬ φ))
3	2	con1d 116	. 2 ⊢ (∀x(φ → ∀xφ) → (¬ ∀x ¬ φ → φ))
4	1, 3	syl5bi 208	1 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

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

This theorem is referenced by:  19.9t  1779  19.9hOLD  1781