induction ls; intuition; simpl in *.
destruct (eqd a2 a); subst.
intuition.
congruence.
simpl.
intuition.
Qed.

Lemma removeFirst_in_iff : forall (A : Set)(eqd : eq_dec A)(ls : list A) a1 a2,
In a1 (removeFirst eqd ls a2) ->
In a1 ls.