Theorem pm11.52 30886

Description: Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.52	|- ( E. x E. y ( ph /\ ps ) <-> -. A. x A. y ( ph -> -. ps ) )

Proof of Theorem pm11.52

Step	Hyp	Ref	Expression
1		df-an 371	. . 3 |- ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) )
2	1	2exbii 1645	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x E. y -. ( ph -> -. ps ) )
3		2nalexn 1629	. 2 |- ( -. A. x A. y ( ph -> -. ps ) <-> E. x E. y -. ( ph -> -. ps ) )
4	2, 3	bitr4i 252	1 |- ( E. x E. y ( ph /\ ps ) <-> -. A. x A. y ( ph -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1377   E.wex 1596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597

This theorem is referenced by: (None)