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

Theorem exan 1656
Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
exan.1  |-  ( E. x ph  /\  ps )
Assertion
Ref Expression
exan  |-  E. x
( ph  /\  ps )

Proof of Theorem exan
StepHypRef Expression
1 hbe1 1456 . . . 4  |-  ( E. x ph  ->  A. x E. x ph )
2119.28h 1526 . . 3  |-  ( A. x ( E. x ph  /\  ps )  <->  ( E. x ph  /\  A. x ps ) )
3 exan.1 . . 3  |-  ( E. x ph  /\  ps )
42, 3mpgbi 1413 . 2  |-  ( E. x ph  /\  A. x ps )
5 19.29r 1585 . 2  |-  ( ( E. x ph  /\  A. x ps )  ->  E. x ( ph  /\  ps ) )
64, 5ax-mp 5 1  |-  E. x
( ph  /\  ps )
Colors of variables: wff set class
Syntax hints:    /\ wa 103   A.wal 1314   E.wex 1453
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-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  bm1.3ii  4019  bdbm1.3ii  13016
  Copyright terms: Public domain W3C validator