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Theorem 19.9t 1533
 Description: A closed version of 19.9 1535. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)
Assertion
Ref Expression
19.9t (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))

Proof of Theorem 19.9t
StepHypRef Expression
1 df-nf 1350 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 19.9ht 1532 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
31, 2sylbi 114 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
4 19.8a 1482 . 2 (𝜑 → ∃𝑥𝜑)
53, 4impbid1 130 1 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  19.9d  1551  19.23t  1567  spimt  1624  exdistrfor  1681  sbequi  1720  sbft  1728  vtoclegft  2625  copsexg  3981
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