Theorem bj-19.9htbi 37605

Description: Strengthening 19.9ht 2351 by replacing its consequent with a biconditional (19.9t 2241 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 2351	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
2		19.8a 2218	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
3	1, 2	impbid1 228	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-12 2213

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  bj-hbntbi  37606