Theorem exanaliim 1700

Description: A transformation of quantifiers and logical connectives. The converse holds in classical but not in intuitionistic logic. (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 697	. . 3 |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )
2	1	eximi 1653	. 2 |- ( E. x ( ph /\ -. ps ) -> E. x -. ( ph -> ps ) )
3		exnalim 1699	. 2 |- ( E. x -. ( ph -> ps ) -> -. A. x ( ph -> ps ) )
4	2, 3	syl 14	1 |- ( E. x ( ph /\ -. ps ) -> -. A. x ( ph -> ps ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1400   E.wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514

This theorem is used by:  rexnalim  2539  nssr  3308  nssssr  4362  brprcneu  5688