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Theorem bj-19.9htbi 32819
Description: Strengthening 19.9ht 2181 by replacing its succedent with a biconditional (19.9t 2109 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-19.9htbi (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Proof of Theorem bj-19.9htbi
StepHypRef Expression
1 19.9ht 2181 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
2 19.8a 2090 . 2 (𝜑 → ∃𝑥𝜑)
31, 2impbid1 215 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by:  bj-hbntbi  32820
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