Theorem alinexa 1494

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 624	. . 3 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2	1	albii 1359	. 2 |- ( A. x ( ph -> -. ps ) <-> A. x -. ( ph /\ ps ) )
3		alnex 1388	. 2 |- ( A. x -. ( ph /\ ps ) <-> -. E. x ( ph /\ ps ) )
4	2, 3	bitri 173	1 |- ( A. x ( ph -> -. ps ) <-> -. E. x ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by:  sbnv  1768  ralnex  2316