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Theorem 19.38b 1808
Description: Under a non-freeness hypothesis, the implication 19.38 1806 can be strengthened to an equivalence. See also 19.38a 1807. (Contributed by BJ, 3-Nov-2021.)
Assertion
Ref Expression
19.38b (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Proof of Theorem 19.38b
StepHypRef Expression
1 19.38 1806 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
2 df-nf 1750 . . 3 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
3 exim 1801 . . . 4 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
4 imim2 58 . . . 4 ((∃𝑥𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
53, 4syl5 34 . . 3 ((∃𝑥𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
62, 5sylbi 207 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
71, 6impbid2 216 1 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
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