ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.43 Unicode version

Theorem 19.43 1535
Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.43  |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )

Proof of Theorem 19.43
StepHypRef Expression
1 hbe1 1400 . . . 4  |-  ( E. x ph  ->  A. x E. x ph )
2 hbe1 1400 . . . 4  |-  ( E. x ps  ->  A. x E. x ps )
31, 2hbor 1454 . . 3  |-  ( ( E. x ph  \/  E. x ps )  ->  A. x ( E. x ph  \/  E. x ps ) )
4 19.8a 1498 . . . 4  |-  ( ph  ->  E. x ph )
5 19.8a 1498 . . . 4  |-  ( ps 
->  E. x ps )
64, 5orim12i 686 . . 3  |-  ( (
ph  \/  ps )  ->  ( E. x ph  \/  E. x ps )
)
73, 6exlimih 1500 . 2  |-  ( E. x ( ph  \/  ps )  ->  ( E. x ph  \/  E. x ps ) )
8 orc 643 . . . 4  |-  ( ph  ->  ( ph  \/  ps ) )
98eximi 1507 . . 3  |-  ( E. x ph  ->  E. x
( ph  \/  ps ) )
10 olc 642 . . . 4  |-  ( ps 
->  ( ph  \/  ps ) )
1110eximi 1507 . . 3  |-  ( E. x ps  ->  E. x
( ph  \/  ps ) )
129, 11jaoi 646 . 2  |-  ( ( E. x ph  \/  E. x ps )  ->  E. x ( ph  \/  ps ) )
137, 12impbii 121 1  |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 102    \/ wo 639   E.wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  19.44  1588  19.45  1589  19.34  1590  sborv  1786  r19.43  2485  rexun  3151  unipr  3622  uniun  3627  unopab  3864  dmun  4570  coundi  4850  coundir  4851
  Copyright terms: Public domain W3C validator