Theorem alinexa 1582

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 679	. . 3 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2	1	albii 1446	. 2 |- ( A. x ( ph -> -. ps ) <-> A. x -. ( ph /\ ps ) )
3		alnex 1475	. 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 1329   E.wex 1468

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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie2 1470

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337

This theorem is referenced by:  sbnv  1860  ralnex  2424