Theorem exnalim 1657

Description: One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)

Assertion

Ref	Expression
exnalim	|- ( E. x -. ph -> -. A. x ph )

Proof of Theorem exnalim

Step	Hyp	Ref	Expression
1		alexim 1656	. 2 |- ( A. x ph -> -. E. x -. ph )
2	1	con2i 628	1 |- ( E. x -. ph -> -. A. x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1362   E.wex 1503

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

This theorem is referenced by:  exanaliim  1658  alexnim  1659  nnal  1660  dtru  4592  brprcneu  5547