Theorem exanaliim 1640

Description: A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)

Assertion

Ref	Expression
exanaliim	|- ( E. x ( ph /\ -. ps ) -> -. A. x ( ph -> ps ) )

Proof of Theorem exanaliim

Step	Hyp	Ref	Expression
1		annimim 681	. . 3 |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )
2	1	eximi 1593	. 2 |- ( E. x ( ph /\ -. ps ) -> E. x -. ( ph -> ps ) )
3		exnalim 1639	. 2 |- ( E. x -. ( ph -> ps ) -> -. A. x ( ph -> ps ) )
4	2, 3	syl 14	1 |- ( E. x ( ph /\ -. ps ) -> -. A. x ( ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1346   E.wex 1485

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454

This theorem is referenced by:  rexnalim  2459  nssr  3207  nssssr  4207  brprcneu  5489