Theorem 19.9ht 1641

Description: A closed version of one direction of 19.9 1644. (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 1449	. 2 ⊢ ∀𝑥(𝜑 → 𝜑)
3		19.23ht 1497	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑)))
4	2, 3	mpbii 148	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1449  ax-ie2 1494

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.9t  1642  19.9h  1643  19.9hd  1662