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Let $a,b$ are sets. If $a=b$, then for all set $x$, $a\in x$ iff $b\in x$?

I'm sure this will hold true, but I'm curious how to prove it.

amoogae
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  • What's the definition about $a, b $ and $x$? –  Oct 15 '20 at 03:11
  • @Ramanujan Thanks. They are sets. – amoogae Oct 15 '20 at 03:42
  • I answered, but looking more closely at your question, you are calling this a “definition of equality” which makes me question what variant of first order logic you are working with. If you are working without equality and without equality in the signature and defining equality this way (which is tenable, if messy, if I recall correctly), then of course this just holds by definition. You will probably get a better answer if you specify what your base system of FOL is. – spaceisdarkgreen Oct 15 '20 at 05:27
  • It more looks like class equality. – zkutch Oct 15 '20 at 08:59
  • шиворот навыворот. – metamorphy Oct 16 '20 at 09:20

1 Answers1

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Yes, this is an instance of the substitution principle and it holds at the level of first order logic (no set theory axioms involved). If $a=b$ then any two formulas are equivalent if the only change is substituting $a$ for $b$.

spaceisdarkgreen
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