How to validate form

How to validate form

Anyone please help me on how to validate a form? I have the code, it is
working ok..but I found that after I hit the submit button & php shows the
errors, there is an input field, asking the user to answer the math
Ques..when I enter the answer, then hit submit button again, it just says
mail successfully sent, but other fields are still empty. Any ideas how to
fix it. Also, the php value in the answerbox input tags, is not saving the
text in box, it disappears. by the way, I got some help from other users,
i'm a noob in php, so if you don't mind please explain.
Thanks for your time!
<form name="contact" action="formtest.php" method="post">
<label for="YourName">Your Name:</label>
<input type="text" name="name" class="required" value="<?=
isset($_POST['name']) ? $_POST['name'] : '' ?>" />
<label for="YourEmail">Your Email:</label>
<input type="text" name="email" class="required" value="<?=
isset($_POST['email']) ? $_POST['email'] : '' ?>"/>
<label for="Subject">Subject:</label>
<input type="text" name="subject" class="required" value="<?=
isset($_POST['subject']) ? $_POST['subject'] : '' ?>" />
<label for="YourMessage">Your Message:</label>
<textarea name="message" class="required"><?= isset($_POST['message'])
? $_POST['message'] : '' ?></textarea>
<p class="c3">10 + 5 =<input type="text" name="answerbox" id="answerbox"
"<?= isset($_POST['answerbox']) ? $_POST['answerbox'] : '' ?>"/></p>
<fieldset>
<input type="submit" name="submit" id="submit" value="Send"
class="required"/>
<input type="reset" id="reset" value="Reset"/>
</fieldset>
<?php
if(isset($_POST['submit'])){
$name = trim($_POST["name"]);
$email = trim($_POST["email"]);
$subject = trim($_POST["subject"]);
$message = trim($_POST["message"]);
$answerbox = trim($_POST["answerbox"]);
if(empty($_POST['name'])){
print "<div class='formerrors'><li>Please type your
name.</li></div>";
}
else {
if (ctype_alpha($name) === false) {
print "<div class='formerrors'><li>Your name only should be in
letters!</li></div>";
}
}
if(empty($_POST['email'])){
print "<div class='formerrors'><li>You've forgot to type your
email address.</li></div>";
} else{
if(filter_var($email, FILTER_VALIDATE_EMAIL) === false){
print "<div class='formerrors'><li>Your email is not valid,
please check.</li></div>";
}
}
if(empty($_POST['subject'])){
print "<div class='formerrors'><li>Please type a
subject.</li></div>";
}
if(empty($_POST['message'])){
print "<div class='formerrors'><li>You've forgot to type your
message.</li></div>";
}
if(empty($_POST['answerbox'])){
print "<div class='formerrors'><li>Please answer the math
question.</li></div>";
}
else {
if($answerbox != 15){
print"<div class='formerrors'><li>Answer is not
correct.</li></div>";
}
else{
$headers = 'From: '.$email. "\r\n" .
'Reply-To: '.$email . "\r\n" .
'X-Mailer: PHP/' . phpversion();
mail('me@mymail.me',$subject,$message,$headers);
print "<div class='formerrors'><li>mail succesuffully
sent</li></div>";
}
}
}
?>
</form>