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Full Idea
The identity of indiscernibles (∀x∀y(∀X(Xx↔Xy)→x=y) is necessarily true, provided that we construe 'property' very broadly, so that 'being a member of such-and-such set' counts as a property.
Gist of Idea
The identity of indiscernibles is necessarily true, if being a member of some set counts as a property
Source
Theodore Sider (Logic for Philosophy [2010], 5.4.3)
Book Ref
Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.125
A Reaction
Sider's example is that if the two objects are the same they must both have the property of being a member of the same singleton set, which they couldn't have if they were different.