Is this alternative definition of 'equivalence relation' well-known? useful? used?

Question

I discovered that $$R \textrm{ is an equivalence relation on } A \;\equiv\; \langle \forall a,b \in A :: aRb \:\equiv\: \langle \forall x \in A :: aRx \equiv bRx\rangle \rangle$$

The nice thing about this alternative definition is that it says, "equivalence of $a$ and $b$ means that it doesn't matter which of the two you substitute", which seems to be the essence of equivalence (cf. Leibniz' law).

Is this alternative definition well-known? useful? used? Update: Or does anyone perhaps have a literature reference?

(I originally asked this same question here: Is this alternative definition of 'equivalence relation' correct? But that question turned into a correctness discussion. Therefore I have posted this as a new question.)

It took me a while to convince myself that your characterization is correct, but it looks good to me. Nice! — Cheerful Parsnip, Jun 25 '13 at 01:25
@Marnix Klooster Could you please add a proof of this nice property ? — , Apr 10 '19 at 21:55
@EleonoreSaintJames That proof is mostly in the original question I point to, https://math.stackexchange.com/q/424576/11994. — MarnixKlooster ReinstateMonica, Apr 11 '19 at 04:10

Brian Rushton · Accepted Answer · 2013-06-29T03:18:17.633

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This equivalence was known at least by 1991, although likely beforehand:http://www.mathmeth.com/tom/files/equivalence.pdf

The author of this note calls it a 'beautiful characterization', and I agree with both of you that it's very cool!

edited Jun 29 '13 at 03:18

answered Jun 26 '13 at 01:41

Brian Rushton

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Thanks for this! And it is rather fitting that this is from someone from the "Dijkstra school", both figuratively and literally, i.e., from the Department of Mathematics and Computing Science at Eindhoven University of Technology. :-) – MarnixKlooster ReinstateMonica Jul 01 '13 at 07:12

Is this alternative definition of 'equivalence relation' well-known? useful? used?

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