Theorem alexnim 1521

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

Assertion

Ref	Expression
alexnim	|- (A.xE.y -. ph -> -. E.xA.yph)

Proof of Theorem alexnim

Step	Hyp	Ref	Expression
1		exnalim 1519	. . 3 |- (E.y -. ph -> -. A.yph)
2	1	alimi 1324	. 2 |- (A.xE.y -. ph -> A.x -. A.yph)
3		alnex 1369	. 2 |- (A.x -. A.yph <-> -. E.xA.yph)
4	2, 3	sylib 127	1 |- (A.xE.y -. ph -> -. E.xA.yph)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1226  E.wex 1362

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 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330

This theorem is referenced by:  nalset  3861  bj-nalset  7118