I whipped up a brute force method which might be of interest as a reference.

# Crude but effective hunter.

use strict;

my %primes = map { $_ => 1 } ( 2,3,5,7 );

for (10234..98765) {
  my @a = split //;
  
  #   Digit 1 is 1 less than digit 3.
  next unless $a[0] == $a[2]-1;
  
  #   Digit 1 is higher than the sum of digits 4 and 5.
  next unless $a[0] > $a[3]+$a[4];
  
  my @sorted = sort @a;
  my $highest = pop @sorted;
  my $lowest = shift @sorted;
  
  #   Digit 3 is the highest number.
  next unless $a[2] == $highest;
  
  #   Digit 2 is lowest.
  next unless $a[1] == $lowest;
  
  #   Digit 5 is between digit 2 and digit 1 and is half of digit 4.
  my ($low, $high) = sort ($a[0], $a[1]);
  next unless $a[4] < $high;
  next unless $a[4] > $low;
  next unless $a[4] == $a[3]/2;
  
  #   It has 2 prime digits.
  my $p_count = 0;
  foreach (@a) {
    $p_count++ if $primes{ $_ };
  }
  next unless $p_count == 2;
  
  #   There are no duplicates.
  my %uniq = ();
  foreach (@a) {
    $uniq{$_} = 1;
  }
  next unless scalar( keys %uniq ) == 5;

  # All tests passed - print number.
  print @a, "\n";
}
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The results produced match those given above - so it must be right....

-- tidiness is the memory loss of environmental mnemonics