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Theorem 19.23t 2211
Description: Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2212. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1786 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 1842 . 2 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
2 19.3t 2202 . . 3 (Ⅎ𝑥𝜓 → (∀𝑥𝜓𝜓))
32imbi2d 344 . 2 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑𝜓)))
41, 3bitr3d 284 1 (Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786
This theorem is referenced by:  19.23  2212  axie2  2789  r19.23t  3299  ceqsalt  3502  vtoclgft  3528  vtoclgftOLD  3529  sbciegft  3783  bj-ceqsalt0  34285  bj-ceqsalt1  34286  wl-equsald  34906
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