Theorem exnalim 1626

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 1625	. 2 |- ( A. x ph -> -. E. x -. ph )
2	1	con2i 617	1 |- ( E. x -. ph -> -. A. x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1330   E.wex 1469

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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438

This theorem is referenced by:  exanaliim  1627  alexnim  1628  dtru  4483  brprcneu  5422  bj-nnal  13120