The Coq Standard Library

# Library Coq.Logic.Classical_Pred_Type

Classical Predicate Logic on Type
Require Import Classical_Prop.
Section Generic.
Variable U : Type.
de Morgan laws for quantifiers
Lemma not_all_not_ex :
forall P:U -> Prop, ~ (forall n:U, ~ P n) -> exists n : U, P n.
Lemma not_all_ex_not :
forall P:U -> Prop, ~ (forall n:U, P n) -> exists n : U, ~ P n.
Lemma not_ex_all_not :
forall P:U -> Prop, ~ (exists n : U, P n) -> forall n:U, ~ P n.
Lemma not_ex_not_all :
forall P:U -> Prop, ~ (exists n : U, ~ P n) -> forall n:U, P n.
Lemma ex_not_not_all :
forall P:U -> Prop, (exists n : U, ~ P n) -> ~ (forall n:U, P n).
Lemma all_not_not_ex :
forall P:U -> Prop, (forall n:U, ~ P n) -> ~ (exists n : U, P n).
End Generic.