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Theorem 19.38a 1833
 Description: Under a non-freeness hypothesis, the implication 19.38 1832 can be strengthened to an equivalence. See also 19.38b 1834. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)
Assertion
Ref Expression
19.38a (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Proof of Theorem 19.38a
StepHypRef Expression
1 19.38 1832 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
2 id 22 . . . 4 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
32nfrd 1785 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4 alim 1804 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
53, 4syl9 77 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
61, 5impbid2 227 1 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1528  ∃wex 1773  Ⅎwnf 1777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803 This theorem depends on definitions:  df-bi 208  df-ex 1774  df-nf 1778 This theorem is referenced by:  19.21t  2199
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