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Theorem nnral 2467
Description: The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1649. (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 2466 . . 3  |-  ( E. x  e.  A  -.  ph 
->  -.  A. x  e.  A  ph )
21con3i 632 . 2  |-  ( -. 
-.  A. x  e.  A  ph 
->  -.  E. x  e.  A  -.  ph )
3 ralnex 2465 . 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 2455   E.wrex 2456
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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-ral 2460  df-rex 2461
This theorem is referenced by:  onntri13  7230  onntri24  7234
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