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Theorem 19.23t 2200
Description: Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2201. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1776 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 8-Oct-2022.)
Assertion
Ref Expression
19.23t (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Proof of Theorem 19.23t
StepHypRef Expression
1 19.38b 1832 . 2 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
2 19.3t 2191 . . 3 (Ⅎ𝑥𝜓 → (∀𝑥𝜓𝜓))
32imbi2d 342 . 2 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑𝜓)))
41, 3bitr3d 282 1 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526  wex 1771  wnf 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-ex 1772  df-nf 1776
This theorem is referenced by:  19.23  2201  axie2  2783  r19.23t  3310  ceqsalt  3525  vtoclgft  3551  vtoclgftOLD  3552  sbciegft  3805  bj-ceqsalt0  34097  bj-ceqsalt1  34098  wl-equsald  34660
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