Theorem alexnim 1539

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 1537	. . 3 |- ( E. y -. ph -> -. A. y ph )
2	1	alimi 1344	. 2 |- ( A. x E. y -. ph -> A. x -. A. y ph )
3		alnex 1388	. 2 |- ( A. x -. A. y ph <-> -. E. x A. y ph )
4	2, 3	sylib 127	1 |- ( A. x E. y -. ph -> -. E. x A. y ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350

This theorem is referenced by:  nalset  3887  bj-nalset  10015