No. If you didn't deliver the title to the dealer at the time of your trade, what probably happened is that included in the pile of documents you signed at the dealer was a form that indicated you did not have the title document and authorized the dealer to obtain a replacement title document in the dealer's name. This effectively negated the title that you still have in your possession.
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The information given here is not legal advice. As all states have different intricacies in their laws, the information given is general only. This communication does not establish an attorney-client relationship with you. I hope this answer has been helpful to you.
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