Theorem exnalim 1577

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 1576	. 2 |- ( A. x ph -> -. E. x -. ph )
2	1	con2i 589	1 |- ( E. x -. ph -> -. A. x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390

This theorem is referenced by:  exanaliim  1578  alexnim  1579  dtru  4303  brprcneu  5191