Theorem alexnim 1696

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 1694	. . 3 |- ( E. y -. ph -> -. A. y ph )
2	1	alimi 1503	. 2 |- ( A. x E. y -. ph -> A. x -. A. y ph )
3		alnex 1547	. 2 |- ( A. x -. A. y ph <-> -. E. x A. y ph )
4	2, 3	sylib 122	1 |- ( A. x E. y -. ph -> -. E. x A. y ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1395   E.wex 1540

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 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509

This theorem is referenced by:  nalset  4219  bj-nalset  16516