Php form validate and post are working but not tog

In order to work properly my form needs to be set up to first check the validation and then post the data if the validation passes. This is fine but I am not sure how to combine the validation code with the post code in the form action. Example if the action is: action="<?php echo htmlspecialchars($_SERVER["PHP_SELF"]);?>"> the form validates correctly but does not send anywhere! If I change the form action to action="contact-engine.php"> then the form is posted but without validation! With this in mind I need to combine into the action both the validation and then (once passed validation) the contact-engine.php problem is I simply do not know how to do this? I really am a learner in php and this is complicated for me! Any help is really appreciated I have been working on this from now for a few days! (N.B. both pages are .php) Full code is as follows:

    <!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="utf-8">
    <title>Help!</title>

    <style type="text/css">
        .error {color: #FF0000;}
    </style>
</head>

<body>

<?php
    // define variables and set to empty values
    $nameErr = $emailErr = "";
    $name = $email = "";

if ($_SERVER["REQUEST_METHOD"] == "POST")
{

//Name
    if (empty($_POST["name"]))
    {$nameErr = "Name is required";}
    else
    {
    $name = test_input($_POST["name"]);
    if (!preg_match("/^[a-zA-Z ]*$/",$name))
    {
    $nameErr = "Only letters and white space allowed";
    }}

//Email
   if (empty($_POST["email"]))
   {$emailErr = "Email is required";}
   else
   {
   $email = test_input($_POST["email"]);
   if (!preg_match("/([\w\-]+\@[\w\-]+\.[\w\-]+)/",$email))
   {
   $emailErr = "Invalid email format";
   }}
}
function test_input($data)
{
     $data = trim($data);
     $data = stripslashes($data);
     $data = htmlspecialchars($data);
     return $data;
}
?>

<form method="post" id="form" action="<?php echo htmlspecialchars($_SERVER["PHP_SELF"]);?>">
    <p><span class="error">* required field.</span></p><br />

    <div class="contact-font" style=" margin-top: 20px;">
        <span class="asterix">* </span>Name:<br />
        <input type="text" name="name" class="border" size="25" value="<?php if(isset($_POST['name'])) {echo $_POST['name']; } ?>">
        <span class="error"><?php echo $nameErr;?></span>
    </div>

    <div class="contact-font" style=" margin-top: 20px;">
        <span class="asterix">* </span>Email: (please double check enty)<br />
        <input type="text" name="email" class="border" size="25" value="<?php if(isset($_POST['email'])) {echo $_POST['email']; } ?>"><span class="error">
        <?php echo $emailErr;?></span>
    </div>

    <div>
        <input type="submit" value="Send" id="submit">
    </div>
</form>

</body>
</html>


And below is the contact-engine code:
<html>
<head>
    <title>Contact Engine</title>
</head>

<body>

    <br />Name:<?php echo htmlspecialchars($_POST['name']); ?><br />
    <br />Email:<?php echo htmlspecialchars($_POST['email']); ?><br />

</body>
</html>

You can try this. This is a simple code below through which you can achieve this:

//sample index.php file
<?php
  include 'submitted.php';
?>
<!DOCTYPE html>
<html>
<head>
</head>
<body>
<form action="<?php echo $_SERVER['PHP_SELF']?>" method="POST">
      <input type="text" name="foo" />
      <input type="submit" value="submit">
</form>
</body>
</html>

Here, I am submitting the form to the same page using post method. But, I have included another file which is include 'submitted.php'.

Here is the test script inside the submitted.php

//sample submitted.php
<?php
  if(isset($_POST['foo']))
 {
    if(strlen($_POST['foo']) < 5){
  echo "String length too small";
  }
  else
  {
   echo $_POST['foo'];
  }
 }
?>

For test, it simply checks if the length is more than five or not. If it is not, the error message is displayed on the same page where your page is present.

Test it yourself too.