Theorem exnalim 1537

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 1536	. 2 |- ( A. x ph -> -. E. x -. ph )
2	1	con2i 557	1 |- ( E. x -. ph -> -. A. x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350

This theorem is referenced by:  exanaliim  1538  alexnim  1539  dtru  4284  brprcneu  5171