Section PredExt_RelChoice_imp_EM.
(** The axiom of extensionality for predicates *)
Definition PredicateExtensionality :=
forall P Q:bool -> Prop, (forall b:bool, P b <-> Q b) -> P = Q.
(** From predicate extensionality we get propositional extensionality hence proof-irrelevance *)
Variable pred_extensionality : PredicateExtensionality.
(** From proof-irrelevance and relational choice, we get guarded relational choice *)
Variable rel_choice : RelationalChoice.
(** The form of choice we need: there is a functional relation which chooses an element in any non empty subset of bool *)

Lemma AC_bool_subset_to_bool :
exists R : (bool -> Prop) -> bool -> Prop,
(forall P:bool -> Prop,
(exists b : bool, P b) ->
exists b : bool, P b /\ R P b /\ (forall b':bool, R P b' -> b = b')).