ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.9t GIF version

Theorem 19.9t 1642
Description: A closed version of 19.9 1644. (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 1461 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 19.9ht 1641 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
31, 2sylbi 121 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
4 19.8a 1590 . 2 (𝜑 → ∃𝑥𝜑)
53, 4impbid1 142 1 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351  wnf 1460  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  19.9d  1661  19.23t  1677  spimt  1736  exdistrfor  1800  sbequi  1839  sbft  1848  vtoclegft  2809  copsexg  4243
  Copyright terms: Public domain W3C validator