Theorem bj-19.9htbi 34037

Description: Strengthening 19.9ht 2339 by replacing its succedent with a biconditional (19.9t 2204 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 2339	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
2		19.8a 2180	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbid1 227	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785

This theorem is referenced by:  bj-hbntbi  34038