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Theorem 19.21t 2059
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2060. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.)
Assertion
Ref Expression
19.21t (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21t
StepHypRef Expression
1 nf5r 2050 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2 alim 1728 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
31, 2syl9 74 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
4 19.9t 2057 . . . 4 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
54imbi1d 329 . . 3 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
6 19.38 1756 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
75, 6syl6bir 242 . 2 (Ⅎ𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓)))
83, 7impbid 200 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700
This theorem is referenced by:  19.21  2060  stdpc5OLD  2062  19.23t  2064  sbal1  2444  sbal2  2445  r19.21t  2934  ceqsalt  3197  sbciegft  3429  bj-ceqsalt0  31867  bj-ceqsalt1  31868  wl-sbhbt  32314  wl-2sb6d  32320  wl-sbalnae  32324  ax12indalem  33048  ax12inda2ALT  33049
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