Theorem alexnim 1579

Description: A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
alexnim	|- ( A. x E. y -. ph -> -. E. x A. y ph )

Proof of Theorem alexnim

Step	Hyp	Ref	Expression
1		exnalim 1577	. . 3 |- ( E. y -. ph -> -. A. y ph )
2	1	alimi 1384	. 2 |- ( A. x E. y -. ph -> A. x -. A. y ph )
3		alnex 1428	. 2 |- ( A. x -. A. y ph <-> -. E. x A. y ph )
4	2, 3	sylib 120	1 |- ( A. x E. y -. ph -> -. E. x A. y ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390

This theorem is referenced by:  nalset  3908  bj-nalset  10686