Form not submitted after a second call

Question

I try to submit a form if a condition is not validated.

$("#support_form").submit(function() {
  if($("#id_reward").attr('value')=="" && $('.reward_item.none.selected').length==0) {
    alert('problem');
    return false;
  } else {
    alert('valid');
    return true;
  }
});

If my form is valid on the first call, my form is well submitted (redirection to another page)

If my form is not valid the first time, and I correct it then, I see the alert box "valid", but my form is not submitted.

Do you have an idea about this problem ?