Theorem 19.9ht 2305

Description: A closed version of 19.9h 2274. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)

Assertion

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

Proof of Theorem 19.9ht

Step	Hyp	Ref	Expression
1		nf5-1 2133	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
2	1	19.9d 2188	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778

This theorem is referenced by:  bj-19.9htbi  36082