Of course not! If the Emancipation Proclamation freed the slaves, why would we have needed the 13th Amendment?
In any event, supposing one is opposed to slavery, a Constitutional amendment is a good idea simply to cement Abolition into fundamental law. While there is little doubt about the moral, political and diplomatic soundness of the Proclamation, it was of dubious legality.
The 13th amendment abolished slavery for most slaves but allowed slavery as punishment for a crime. It established the double sentencing declarations still in practice today where the punishment is a time and a fine, i.e., 60 dollars or 60 days. The criminal was not afforded the opportunity to pay, he had to serve the time. Factories, sweatshops and businesses needing cheap labor would pay the fine to the court and the criminal would then be owned for the time period involved and all he got was room and board. The practice was gradually abandoned in the early 1900s due to pressure from labor unions who complained that the system was taking the work away from union members.
This sort of offers a solution for today’s prison overcrowding problem. Let’s have an auction.
The main effect of the 13th Amendment was to free slaves in states that didn't join the Confederacy, e.g. Kentucky.
So it appears, that by simply skipping that issue, leaving it up to individual States to put what privileges and immunities they wanted into their Constitution, the Union was formed. But still the work was not done, since for instance, the most obvious short coming, the BOR was not explicitly forced to be recognized by the States. Hence the line of thinking that the reconstruction amendments are of unique importance and I think in many ways they do constitute a second founding of the country. Lincoln had long promised a “new birth of freedom”, and this is likely what that meant.
It looks like the current Chicago gun rights case may prove this out. If this view prevails, it allows for a more expansive view of privileges and immunities with respect to what may or may not be Constitutional.
To end slavery altogether. The Emancipation Proclamation freed the slaves but it didn't outlaw the institution completely. Constitutionally it couldn't.