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Theorem nnral 2484
Description: The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1660. (Contributed by Jim Kingdon, 1-Aug-2024.)
Assertion
Ref Expression
nnral  |-  ( -. 
-.  A. x  e.  A  ph 
->  A. x  e.  A  -.  -.  ph )

Proof of Theorem nnral
StepHypRef Expression
1 rexnalim 2483 . . 3  |-  ( E. x  e.  A  -.  ph 
->  -.  A. x  e.  A  ph )
21con3i 633 . 2  |-  ( -. 
-.  A. x  e.  A  ph 
->  -.  E. x  e.  A  -.  ph )
3 ralnex 2482 . 2  |-  ( A. x  e.  A  -.  -.  ph  <->  -.  E. x  e.  A  -.  ph )
42, 3sylibr 134 1  |-  ( -. 
-.  A. x  e.  A  ph 
->  A. x  e.  A  -.  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wral 2472   E.wrex 2473
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-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-ral 2477  df-rex 2478
This theorem is referenced by:  onntri13  7288  onntri24  7292
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