Axiom talk:Leibniz's Law

Leibniz's Law should not be adopted as an axiom, but rather can be proven (in set theory) from the definition of set equality and induction on the well-formed parts of $P(x)$. One direction is Substitutivity of Equality, while the other direction is a special case of Definition:Set Equality. --Andrew Salmon 23:12, 6 August 2012 (UTC)
 * Are you sure the links you are pointing to are talking about the equality of $x$ and $y$, rather than the equality of $P$ to another $P$ ? --GFauxPas 04:33, 7 August 2012 (UTC)
 * Yes. --Andrew Salmon 04:36, 7 August 2012 (UTC)
 * Can you finish Substitutivity of Equality so I can think about your comment more? In any event, it needs to be kept as an axiom for systems where it can't be proven in. --GFauxPas 04:44, 7 August 2012 (UTC)