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Yeah, it's not a theorem. In math, you start with axioms or postulates. These are assumed to be true without proof. Transitivity is assumed to hold for equality. In general an equivalence relation is one that has these properties:[list=1][*]Reflexive  a=a[*]Symmetric  If a=b, then b=a[*]Transitive  If a=b and b=c, then a=c[/list=1]
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That's not a tau neutrino in my pocket; I've got a hadron. 




So for instance, for people, "lives in the same country as" would be an equivalence relation.
"Is at least as tall as" would not be an equivalence relation because it's not symmetric. "Has been in the same room as" would not be an equivalence relation because it is not transitive. 




Actually, the transitive property of equality *is* a theorem. If you have reflexivity (for all a, a = a) and the principle of substitution (if t = s and P(t) is a statement about t, then P(s) has the same truth value as P(t)), you can derive symmetry (for all a and b, if a = b then b = a) and transitivity (as in the OP).
But this is an advanced matter. 99% of the mathematicians out there regard it as a postulate. 




Worse, ultrafilter's definition of the principle of substitution uses the equality of t=s, which as DrMatrix mentioned is usually defined by using the idea of transitivity. So, it'd be tough to use that to derive transitivity.





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David Simmons: yes. It's an algebraic equivalent of the first General Axiom from Euclid's 'The Elements'. To Euclid, it was one of a number of selfevident notions on which further his further reasoning was based.





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Messy? You don't know the half of it.
What I should've written is this: For any binary relation R, if R(a, a) for all a, and R(t, s) implies that P(t) has the same truth value as P(s) for any statement P, then R(a, b) implies R(b, a) for all a, b, and R(a, b) and R(b, c) implies R(a, c) for all a, b, c. In such case, R has all the same properties as '=', so that's how we usually denote it. We do use '=' to represent weaker relationsfor instance, most equivalence relations do not have the principle of substitution. That's why this is not just a rearrangement of axioms into theorems and vice versa. You can have reflexivity, transitivity, and symmetry without being able to substitute. Note that for any particular area, the concept of 'a = b' is not undefined, but the symbols are undefined until we're talking about a particular area. Is this a little clearer? 




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