Theorem pm11.52 36372

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 372	. . 3 |- ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) )
2	1	2exbii 1715	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x E. y -. ( ph -> -. ps ) )
3		2nalexn 1696	. 2 |- ( -. A. x A. y ( ph -> -. ps ) <-> E. x E. y -. ( ph -> -. ps ) )
4	2, 3	bitr4i 255	1 |- ( E. x E. y ( ph /\ ps ) <-> -. A. x A. y ( ph -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   E.wex 1659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660

This theorem is referenced by: (None)