Theorem 2nalexn 1755

Description: Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
2nalexn	|- ( -. A. x A. y ph <-> E. x E. y -. ph )

Proof of Theorem 2nalexn

Step	Hyp	Ref	Expression
1		df-ex 1705	. . 3 |- ( E. x E. y -. ph <-> -. A. x -. E. y -. ph )
2		alex 1753	. . . 4 |- ( A. y ph <-> -. E. y -. ph )
3	2	albii 1747	. . 3 |- ( A. x A. y ph <-> A. x -. E. y -. ph )
4	1, 3	xchbinxr 325	. 2 |- ( E. x E. y -. ph <-> -. A. x A. y ph )
5	4	bicomi 214	1 |- ( -. A. x A. y ph <-> E. x E. 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:  spc2gv  3296  hashfun  13224  spc2d  29313  pm11.52  38586  2exanali  38587