Theorem alinexa 1591

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)

Assertion

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

Proof of Theorem alinexa

Step	Hyp	Ref	Expression
1		imnan 680	. . 3 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2	1	albii 1458	. 2 |- ( A. x ( ph -> -. ps ) <-> A. x -. ( ph /\ ps ) )
3		alnex 1487	. 2 |- ( A. x -. ( ph /\ ps ) <-> -. E. x ( ph /\ ps ) )
4	2, 3	bitri 183	1 |- ( A. x ( ph -> -. ps ) <-> -. E. x ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie2 1482

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349

This theorem is referenced by:  sbnv  1876  ralnex  2454