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Theorem 19.21t 2171
Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2172. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1766 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 1821 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
2 19.9t 2169 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
32imbi1d 343 . 2 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
41, 3bitr3d 282 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1520  wex 1761  wnf 1765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-12 2141
This theorem depends on definitions:  df-bi 208  df-ex 1762  df-nf 1766
This theorem is referenced by:  19.21  2172  sbal1  2525  sbal2  2526  sbal2OLD  2527  sbal2OLDOLD  2528  r19.21t  3181  ceqsalt  3470  sbciegft  3740  bj-ceqsalt0  33783  bj-ceqsalt1  33784  wl-sbhbt  34347  wl-2sb6d  34351  wl-sbalnae  34355  wl-dfralf  34396  ax12indalem  35638  ax12inda2ALT  35639
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