assert (H:forall p:Prop, (¬p) ≡≡ (p ≡≡ F)).
apply T5.
specialize (H (¬p)).
assert (H1:((¬p ≡≡ p) ≡≡ F)).
apply (contradict p).
assert (H2:((¬p ≡≡ p) ≡≡ F) ≡≡ (F ≡≡ (¬p ≡≡ p))).
apply (T4 (¬p ≡≡ p) F).
apply (detachL ((¬p ≡≡ p) ≡≡ F)(F ≡≡ (¬p ≡≡ p))) in H2;[
clear H1;assert (H1:(F ≡≡ (¬p ≡≡ p)) ≡≡ ((F ≡≡ ¬p) ≡≡ p));[
apply (A2 F (¬p) p) |
apply (detachL (F ≡≡ (¬p ≡≡ p))((F ≡≡ ¬p) ≡≡ p)) in H1;[
clear H2;assert (H2:(F ≡≡ ¬p) ≡≡ ((¬p) ≡≡ F));[
apply (T4 F (¬p)) |
rewrite H2 in H1;clear H2;assert (H2:((¬p ≡≡ F) ≡≡ p) ≡≡ (¬(¬p) ≡≡ (¬p ≡≡ F)));[
apply (intro_eqP ((¬p ≡≡ F) ≡≡ p)(¬(¬p) ≡≡ (¬p ≡≡ F)) H1 H) |
assert (H3:(((¬p ≡≡ F) ≡≡ p) ≡≡ (¬(¬p) ≡≡ (¬p ≡≡ F))) ≡≡ (p ≡≡ ¬(¬p)));[
apply (A1 (¬p ≡≡ F)(¬(¬p)) p) |
apply (detachL (((¬p ≡≡ F) ≡≡ p) ≡≡ (¬(¬p) ≡≡ (¬p ≡≡ F)))(p ≡≡ ¬(¬p))) in H3;[
clear H2;assert (H2:(p ≡≡ ¬(¬p)) ≡≡ (¬(¬p) ≡≡ p));[
apply (T4 p (¬(¬p))) |
apply (detachL (p ≡≡ ¬(¬p))(¬(¬p) ≡≡ p)) in H2;assumption ] |
assumption ]]]] |
assumption ]] |
assumption ].
Qed.

(**** elimination of double negation *****)
Lemma elim_dneg   : forall (p:Prop), (¬¬p) -> p.