Theorem nnal 1697

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

Step	Hyp	Ref	Expression
1		exnalim 1694	. . 3 |- ( E. x -. ph -> -. A. x ph )
2	1	con3i 637	. 2 |- ( -. -. A. x ph -> -. E. x -. ph )
3		alnex 1547	. 2 |- ( A. x -. -. ph <-> -. E. x -. ph )
4	2, 3	sylibr 134	1 |- ( -. -. A. x ph -> A. x -. -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1395   E.wex 1540

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 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509

This theorem is referenced by:  bj-stal  16345