FYI the Constitution doesn't say that. The US declaration of independence does.

It doesn't. The Declaration of Independence says "all men are created equal," but the Constitution does not. The Constitution is much more of a legal document outlining how the government was to work. The document was also a compromise that was made understanding slavery was an issue. If it banned slavery, it wouldn't have gotten ratified. But the issue is clearly there based on the way they dance around it. Nowhere is the word slavery ever mentioned. The Fugitive slave clause just says "person held to service or labour in one state, under the laws thereof." The 3/5 clause goes even farther out of its way. It refers to slaves essentially as "anyone else we haven't mentioned previously" as if it was an afterthought and not the core component of the whole framing of the provision. There's also the clause designed to allow slaves to be imported until at least 1808 is just referred to as "migration or importation of such persons as the states ... think proper to admit," which could make it seem like a reference to immigration. So it's clear the Constitution did not provide for the abolition of slavery (in fact, if you want, I could go into detail about English legal theory that justified the existence of slavery), but it was still the Elephant in the room.

However, the ideals expressed in the Declaration of Independence were certainly influential. I don't think it's a coincidence that so many northern states (where slavery wasn't essential to the economy) provided for gradual abolition right after the American Revolution. It's also why John C. Calhoun, the main southern intellectual thinker in defense of slavery, fought so hard to essentially discredit the concept of the Declaration of Independence. Here's part of one speech:

If he should possess a philosophical turn of mind, and be disposed to look at more remote and recondite causes, he will trace it to a proposition which originated in a hypothetical truism, but which, as now expressed and now understood, is the most false and dangerous of all political error. The proposition to which I allude has become an axiom in the minds of a vast majority on both sides of the Atlantic, and is repeated daily, from tongue to tongue, as an established and incontrovertible truth; it is, that “all men are born free and equal.” I am not afraid to attack error, however deeply it may be intrenched, or however widely extended, whenever it becomes my duty to do so, as I believe it to be on this subject and occasion.

Taking the proposition literally, (it is in that sense it is understood,) there is not a word of truth in it. It begins with “all men are born,” which is utterly untrue. Men are not born. Infants are born. They grow to be men. And concludes with asserting that they are born “free and equal,” which is not less false. They are not born free. While infants they are incapable of freedom, being destitute alike of the capacity of thinking and acting, without which there can be no freedom. Besides, they are necessarily born subject to their parents, and remain so among all people, savage and civilized, until the development of their intellect and physical capacity enables them to take care of themselves. They grow to all the freedom, of which the condition in which they were born permits, by growing to be men. Nor is it less false that they are born “equal.” They are not so in any sense in which it can be regarded; and thus, as I have asserted, there is not a word of truth in the whole proposition, as expressed and generally understood.

http://www.indiana.edu/~kdhist/H105-documents-web/week13/CalhounreOregon1848.html