Theorem ralinexa 2405

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 659	. . 3 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2	1	ralbii 2384	. 2 |- ( A. x e. A ( ph -> -. ps ) <-> A. x e. A -. ( ph /\ ps ) )
3		ralnex 2369	. 2 |- ( A. x e. A -. ( ph /\ ps ) <-> -. E. x e. A ( ph /\ ps ) )
4	2, 3	bitri 182	1 |- ( A. x e. A ( ph -> -. ps ) <-> -. E. x e. A ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   A.wral 2359   E.wrex 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie2 1428  ax-4 1445  ax-17 1464

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-ral 2364  df-rex 2365

This theorem is referenced by: (None)