ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnal Unicode version

Theorem nnal 1636
Description: The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
nnal  |-  ( -. 
-.  A. x ph  ->  A. x  -.  -.  ph )

Proof of Theorem nnal
StepHypRef Expression
1 exnalim 1633 . . 3  |-  ( E. x  -.  ph  ->  -. 
A. x ph )
21con3i 622 . 2  |-  ( -. 
-.  A. x ph  ->  -. 
E. x  -.  ph )
3 alnex 1486 . 2  |-  ( A. x  -.  -.  ph  <->  -.  E. x  -.  ph )
42, 3sylibr 133 1  |-  ( -. 
-.  A. x ph  ->  A. x  -.  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1340   E.wex 1479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-fal 1348  df-nf 1448
This theorem is referenced by:  bj-stal  13524
  Copyright terms: Public domain W3C validator