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Theorem 19.38b 1916
Description: Under a non-freeness hypothesis, the implication 19.38 1913 can be strengthened to an equivalence. See also 19.38a 1914. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)
Assertion
Ref Expression
19.38b (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Proof of Theorem 19.38b
StepHypRef Expression
1 19.38 1913 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
2 exim 1908 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
3 id 22 . . . . 5 (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
43nfrd 1864 . . . 4 (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓))
54imim2d 57 . . 3 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
62, 5syl5 34 . 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 1628  wex 1851  wnf 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884
This theorem depends on definitions:  df-bi 197  df-ex 1852  df-nf 1857
This theorem is referenced by: (None)
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