Theorem exanaliim 1670

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 688	. . 3 |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )
2	1	eximi 1623	. 2 |- ( E. x ( ph /\ -. ps ) -> E. x -. ( ph -> ps ) )
3		exnalim 1669	. 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 104   A.wal 1371   E.wex 1515

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484

This theorem is referenced by:  rexnalim  2495  nssr  3253  nssssr  4266  brprcneu  5569