Theorem alexnim 1641

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 1639	. . 3 |- ( E. y -. ph -> -. A. y ph )
2	1	alimi 1448	. 2 |- ( A. x E. y -. ph -> A. x -. A. y ph )
3		alnex 1492	. 2 |- ( A. x -. A. y ph <-> -. E. x A. y ph )
4	2, 3	sylib 121	1 |- ( A. x E. y -. ph -> -. E. x A. y ph )

Syntax hints:   -. wn 3   -> wi 4   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:  nalset  4119  bj-nalset  13930