The following should work for any systems of expressions that aren't cyclic. It allows multiple dependent expressions as shown by the additional g expression.

use strict;
use warnings;
use Graph;

my $file = <<FILE;
e = a & b;
f = e | c;
d = ~f;
g = ~f | e;
FILE

open my $fIn, '<', \$file or die "Can't open file: $!\n";

my %subst;
my $dag = Graph->new();

while (<$fIn>) {
    chomp;
    s/;$//;

    my ($lhs, $rhs) = split /\s*=\s*/, $_, 2;
    my %verticies = map {$_ => 1} $rhs =~ /\b(\w+)\b/g;

    $subst{$lhs} = $rhs;

    for my $vertex (keys %verticies) {
        $dag->add_edge($lhs, $vertex) if ! $dag->has_edge ($lhs, $vert
+ex);
    }
}

die "Expression set contains cycles. Can't process it\n" if ! $dag->is
+_dag();

my %roots = map {$_ => 1} $dag->source_vertices();
my %sinks = map {$_ => 1} $dag->sink_vertices();

while (keys %sinks) {
    my $sink = (keys %sinks)[0];

    delete $sinks{$sink};
    for my $pred ($dag->predecessors($sink)) {
        $sinks{$pred} = 1;
        $subst{$pred} =~ s/\b$sink\b/($subst{$sink})/g if exists $subs
+t{$sink};
    }
}

print "$_ = $subst{$_}\n" for sort keys %roots;
[download]

Prints:

d = ~((a & b) | c)
g = ~((a & b) | c) | (a & b)
[download]