Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-nnal Unicode version

Theorem bj-nnal 12963
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
bj-nnal  |-  ( -. 
-.  A. x ph  ->  A. x  -.  -.  ph )

Proof of Theorem bj-nnal
StepHypRef Expression
1 exnalim 1625 . . 3  |-  ( E. x  -.  ph  ->  -. 
A. x ph )
21con3i 621 . 2  |-  ( -. 
-.  A. x ph  ->  -. 
E. x  -.  ph )
3 alnex 1475 . 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 1329   E.wex 1468
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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437
This theorem is referenced by:  bj-stal  12971
  Copyright terms: Public domain W3C validator