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Theorem 19.38a 1908
 Description: Under a non-freeness hypothesis, the implication 19.38 1907 can be strengthened to an equivalence. See also 19.38b 1909. (Contributed by BJ, 3-Nov-2021.)
Assertion
Ref Expression
19.38a (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Proof of Theorem 19.38a
StepHypRef Expression
1 19.38 1907 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
2 df-nf 1851 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3 alim 1879 . . . 4 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
4 imim1 83 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜑) → ((∀𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
53, 4syl5 34 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
62, 5sylbi 207 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
71, 6impbid2 216 1 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1622  ∃wex 1845  Ⅎwnf 1849 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878 This theorem depends on definitions:  df-bi 197  df-ex 1846  df-nf 1851 This theorem is referenced by:  19.21t  2212
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