How can I draw the link between negation inconsistency and this result?

Question

If Γ ⊢ ∃x∃y (φ(x)∧φ(y)∧x≠y) , how does adding ∀x(x=c↔φ(x)) result in an inconsistent set?

I know that inconsistent has two variants, namely negation inconsistency and absolute inconsistency? How can I prove the former?

Answer 1

The first formula asserts that there are two different objects that are $\varphi$.

The first formula implies that every object that is $\varphi$ is equal to $c$.

From the first one, by Existential instantiation,we get: $\varphi(c_1)$ and $\varphi(c_2)$, with $\lnot (c_1 = c_2)$.

Using the second one, by Universal instantiation, we get: $c_1=c$ and $c_2=c$, from which [transitivity of equality] we have: $c_1=c_2$, and thus the contradiction.