Theorem bj-19.9htbi 36647

Description: Strengthening 19.9ht 2317 by replacing its consequent with a biconditional (19.9t 2200 does have a biconditional consequent). 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 2317	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
2		19.8a 2177	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbid1 225	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1533  ∃wex 1774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-10 2137  ax-12 2173

This theorem depends on definitions:  df-bi 207  df-ex 1775  df-nf 1779

This theorem is referenced by:  bj-hbntbi  36648