Theorem 19.9ht 1529

Description: A closed version of one direction of 19.9 1532. (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		id 19	. . 3 ⊢ (φ → φ)
2	1	ax-gen 1335	. 2 ⊢ ∀x(φ → φ)
3		19.23ht 1383	. 2 ⊢ (∀x(φ → ∀xφ) → (∀x(φ → φ) ↔ (∃xφ → φ)))
4	2, 3	mpbii 136	1 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))

Syntax hints:   → wi 4  ∀wal 1240  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1335  ax-ie2 1380

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.9t  1530  19.9h  1531  19.9hd  1549