Theorem bj-nnal 12952

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

Step	Hyp	Ref	Expression
1		exnalim 1625	. . 3 |- ( E. x -. ph -> -. A. x ph )
2	1	con3i 621	. 2 |- ( -. -. A. x ph -> -. E. x -. ph )
3		alnex 1475	. 2 |- ( A. x -. -. ph <-> -. E. x -. ph )
4	2, 3	sylibr 133	1 |- ( -. -. A. x ph -> A. x -. -. ph )

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  12960