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Theorem 19.21t 2229
 Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2231. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1858 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 1915 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
2 19.9t 2227 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
32imbi1d 330 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
41, 3bitr3d 270 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1629  ∃wex 1852  Ⅎwnf 1856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203 This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858 This theorem is referenced by:  19.21  2231  stdpc5OLD  2233  19.23t  2235  sbal1  2608  sbal2  2609  r19.21t  3104  ceqsalt  3380  sbciegft  3618  bj-ceqsalt0  33202  bj-ceqsalt1  33203  wl-sbhbt  33670  wl-2sb6d  33675  wl-sbalnae  33679  ax12indalem  34753  ax12inda2ALT  34754
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