Theorem ralinexa 2997

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)

Assertion

Ref	Expression
ralinexa	|- ( A. x e. A ( ph -> -. ps ) <-> -. E. x e. A ( ph /\ ps ) )

Proof of Theorem ralinexa

Step	Hyp	Ref	Expression
1		imnan 438	. . 3 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2	1	ralbii 2980	. 2 |- ( A. x e. A ( ph -> -. ps ) <-> A. x e. A -. ( ph /\ ps ) )
3		ralnex 2992	. 2 |- ( A. x e. A -. ( ph /\ ps ) <-> -. E. x e. A ( ph /\ ps ) )
4	2, 3	bitri 264	1 |- ( A. x e. A ( ph -> -. ps ) <-> -. E. x e. A ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wral 2912   E.wrex 2913

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-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  kmlem7  8978  kmlem13  8984  lspsncv0  19146  ntreq0  20881  lhop1lem  23776  soseq  31751  ltrnnid  35422