Theorem alexn 1771

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

Assertion

Ref	Expression
alexn	|- ( A. x E. y -. ph <-> -. E. x A. y ph )

Proof of Theorem alexn

Step	Hyp	Ref	Expression
1		exnal 1754	. . 3 |- ( E. y -. ph <-> -. A. y ph )
2	1	albii 1747	. 2 |- ( A. x E. y -. ph <-> A. x -. A. y ph )
3		alnex 1706	. 2 |- ( A. x -. A. y ph <-> -. E. x A. y ph )
4	2, 3	bitri 264	1 |- ( A. x E. y -. ph <-> -. E. x A. y ph )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  2exnexn  1772  nalset  4795  kmlem2  8973  bj-nalset  32794