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Theorem 19.21t 2195
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2196. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1779 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
Assertion
Ref Expression
19.21t (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21t
StepHypRef Expression
1 19.38a 1835 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
2 19.9t 2193 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
32imbi1d 341 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
41, 3bitr3d 281 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532  wex 1774  wnf 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-ex 1775  df-nf 1779
This theorem is referenced by:  19.21  2196  sbal1  2523  sbal2  2524  r19.21t  3247  ceqsal1t  3502  sbciegftOLD  3815  bj-ceqsalt0  36362  bj-ceqsalt1  36363  wl-sbhbt  37021  wl-2sb6d  37025  wl-sbalnae  37029  ax12indalem  38417  ax12inda2ALT  38418
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