Even more clean

Example.lean

@@ -2,73 +2,64 @@ variable (p q r : Prop)
 
 -- commutativity of ∧ and ∨
 example : p ∧ q ↔ q ∧ p :=
-  Iff.intro (λ ⟨p,q⟩ => ⟨q,p⟩) (λ ⟨q,p⟩ => ⟨p,q⟩)
+  ⟨λ ⟨p,q⟩ => ⟨q,p⟩, λ ⟨q,p⟩ => ⟨p,q⟩⟩
 
 example : p ∨ q ↔ q ∨ p :=
-  Iff.intro
-    (λ
-      | Or.inl p => Or.inr p
-      | Or.inr q => Or.inl q)
-    (λ
-      | Or.inl q => Or.inr q
-      | Or.inr p => Or.inl p)
+  ⟨λ
+    | Or.inl p => Or.inr p
+    | Or.inr q => Or.inl q,
+   λ
+    | Or.inl q => Or.inr q
+    | Or.inr p => Or.inl p⟩
 
 -- associativity of ∧ and ∨
 example : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) :=
-  Iff.intro
-    (λ ⟨⟨p,q⟩,r⟩ => ⟨p,⟨q,r⟩⟩)
-    (λ ⟨p,⟨q,r⟩⟩ => ⟨⟨p,q⟩,r⟩)
+  ⟨λ ⟨⟨p,q⟩,r⟩ => ⟨p,⟨q,r⟩⟩, λ ⟨p,⟨q,r⟩⟩ => ⟨⟨p,q⟩,r⟩⟩
 
 example : (p ∨ q) ∨ r ↔ p ∨ (q ∨ r) :=
-  Iff.intro
-  (λ
+  ⟨λ
     | Or.inr r => Or.inr (Or.inr r)
     | Or.inl (Or.inr q) => Or.inr (Or.inl q)
-    | Or.inl (Or.inl p) => Or.inl p)
-  (λ
+    | Or.inl (Or.inl p) => Or.inl p,
+  λ
     | Or.inl p => Or.inl (Or.inl p)
     | Or.inr (Or.inl q) => Or.inl (Or.inr q)
-    | Or.inr (Or.inr r) => Or.inr r)
+    | Or.inr (Or.inr r) => Or.inr r⟩
 
 -- distributivity
 example : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) :=
-  Iff.intro
-  (λ
+  ⟨λ
     | ⟨p, Or.inl q⟩ => Or.inl ⟨p,q⟩
-    | ⟨p, Or.inr r⟩ => Or.inr ⟨p,r⟩)
-  (λ
+    | ⟨p, Or.inr r⟩ => Or.inr ⟨p,r⟩,
+  λ
     | Or.inl ⟨p,q⟩ => ⟨p,Or.inl q⟩
-    | Or.inr ⟨p,r⟩ => ⟨p,Or.inr r⟩)
+    | Or.inr ⟨p,r⟩ => ⟨p,Or.inr r⟩⟩
 
 example : p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r) :=
-  Iff.intro
-  (λ
+  ⟨λ
     | Or.inl p => ⟨Or.inl p, Or.inl p⟩
-    | Or.inr ⟨q,r⟩ => ⟨Or.inr q, Or.inr r⟩)
-  (λ
+    | Or.inr ⟨q,r⟩ => ⟨Or.inr q, Or.inr r⟩,
+  λ
     | ⟨Or.inr q, Or.inr r⟩ => Or.inr ⟨q,r⟩
     | ⟨Or.inl p,        _⟩ => Or.inl p
-    | ⟨       _, Or.inl p⟩ => Or.inl p)
+    | ⟨       _, Or.inl p⟩ => Or.inl p⟩
 
 -- other properties
 example : (p → (q → r)) ↔ (p ∧ q → r) :=
-  Iff.intro
-    (λ f ⟨p,q⟩ => f p q)
-    (λ f p q => f ⟨p,q⟩)
+  ⟨λ f ⟨p,q⟩ => f p q,λ f p q => f ⟨p,q⟩⟩
 
 example : ((p ∨ q) → r) ↔ (p → r) ∧ (q → r) :=
-  Iff.intro
-    (λ porqtor => And.intro (λ p => porqtor (Or.inl p)) (λ q => porqtor (Or.inr q)))
-    (λ
+  ⟨
+    λ f =>  ⟨λ p => f (Or.inl p), λ q => f (Or.inr q)⟩,
+    λ
       |⟨f, _⟩, Or.inl p => f p
-      |⟨_, g⟩, Or.inr q => g q)
+      |⟨_, g⟩, Or.inr q => g q⟩
 
 example : ¬(p ∨ q) ↔ ¬p ∧ ¬q :=
-  Iff.intro
-    (λ nporq => And.intro (λ p => nporq (Or.inl p)) (λ q => nporq (Or.inr q)))
-    (λ
-      | ⟨np, _⟩, Or.inl p => np p
-      | ⟨_ ,nq⟩, Or.inr q => nq q)
+  ⟨λ f => ⟨λ p => f (Or.inl p), λ q => f (Or.inr q)⟩,
+   λ
+    | ⟨np, _⟩, Or.inl p => np p
+    | ⟨_ ,nq⟩, Or.inr q => nq q⟩
 
 example : ¬p ∨ ¬q → ¬(p ∧ q)
   | Or.inl np, ⟨p,_⟩ => np p
@@ -88,10 +79,10 @@ example : (¬p ∨ q) → (p → q)
   | (Or.inr q), _ => q
 
 example : p ∨ False ↔ p :=
-  Iff.intro (λ porf => Or.elim porf id False.elim) (λ p => Or.inl p)
+  ⟨λ h => Or.elim h id False.elim, λ p => Or.inl p⟩
 
 example : p ∧ False ↔ False :=
-  Iff.intro (λ ⟨_,f⟩ => f) (λ f => False.elim f)
+  ⟨λ ⟨_,f⟩ => f, λ f => False.elim f⟩
 
 example : (p → q) → (¬q → ¬p) :=
- λ ptoq nq p => absurd (ptoq p) nq
+ λ f nq p => absurd (f p) nq