Theorem 19.9ht 2019

Description: A closed version of 19.9 2022. (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		exim 1739	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑))
2		axc7e 1993	. 2 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑)
3	1, 2	syl6 34	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1472  ∃wex 1694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983

This theorem depends on definitions:  df-bi 195  df-ex 1695

This theorem is referenced by:  19.9d  2020  hbnt  2024  bj-19.9htbi  31716