Theorem alexn 1579

Description: A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)

Assertion

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

Proof of Theorem alexn

Step	Hyp	Ref	Expression
1		exnal 1574	. . 3 |- (E.y -. ph <-> -. A.yph)
2	1	albii 1566	. 2 |- (A.xE.y -. ph <-> A.x -. A.yph)
3		alnex 1543	. 2 |- (A.x -. A.yph <-> -. E.xA.yph)
4	2, 3	bitri 240	1 |- (A.xE.y -. ph <-> -. E.xA.yph)

Syntax hints:  -. wn 3   <-> wb 176  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  2exnexn  1580