variant trans_eq : \forall A:Type.\forall x,y,z:A. x=y \to y=z \to x=z
\def transitive_eq.
+theorem symmetric_not_eq: \forall A:Type. symmetric A (λx,y.x ≠ y).
+unfold symmetric.simplify.intros.unfold.intro.apply H.apply sym_eq.assumption.
+qed.
+
+variant sym_neq: ∀A:Type.∀x,y.x ≠ y →y ≠ x
+≝ symmetric_not_eq.
+
theorem eq_elim_r:
\forall A:Type.\forall x:A. \forall P: A \to Prop.
P x \to \forall y:A. y=x \to P y.