Theorem bj-19.9htbi 32694

Description: Strengthening 19.9ht 2143 by replacing its succedent with a biconditional (19.9t 2071 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.)

Assertion

Ref	Expression
bj-19.9htbi	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))

Proof of Theorem bj-19.9htbi

Step	Hyp	Ref	Expression
1		19.9ht 2143	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
2		19.8a 2052	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbid1 215	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-hbntbi  32695