ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nexd Unicode version

Theorem nexd 1549
Description: Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
Hypotheses
Ref Expression
nexd.1  |-  ( ph  ->  A. x ph )
nexd.2  |-  ( ph  ->  -.  ps )
Assertion
Ref Expression
nexd  |-  ( ph  ->  -.  E. x ps )

Proof of Theorem nexd
StepHypRef Expression
1 nexd.1 . . 3  |-  ( ph  ->  A. x ph )
2 nexd.2 . . 3  |-  ( ph  ->  -.  ps )
31, 2alrimih 1403 . 2  |-  ( ph  ->  A. x  -.  ps )
4 alnex 1433 . 2  |-  ( A. x  -.  ps  <->  -.  E. x ps )
53, 4sylib 120 1  |-  ( ph  ->  -.  E. x ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1287   E.wex 1426
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
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295
This theorem is referenced by:  nexdv  1859
  Copyright terms: Public domain W3C validator