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Theorem 19.9t 1574
Description: A closed version of 19.9 1576. (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 1391 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 19.9ht 1573 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
31, 2sylbi 119 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
4 19.8a 1523 . 2 (𝜑 → ∃𝑥𝜑)
53, 4impbid1 140 1 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  wnf 1390  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  19.9d  1592  19.23t  1608  spimt  1665  exdistrfor  1722  sbequi  1761  sbft  1770  vtoclegft  2671  copsexg  4001
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