macro "conj_elim" hp:ident hq:ident "from" h:ident : tactic =>
  `(tactic|(have ⟨$hp,$hq⟩ := $h))

macro "conj_intro" h:ident "from" hp:ident hq:ident : tactic =>
  `(tactic|(have $h : _∧_ := And.intro $hp $hq))

macro "disj_elim" hp:ident hq:ident "from" h:term:arg : tactic =>
  `(tactic|(refine Or.elim $h (fun $hp => ?_) (fun $hq => ?_)))

macro "disj_intro_left" h:ident "from" hp:ident q:term:arg : tactic =>
  `(tactic|(have $h : _∨($q) := Or.inl $hp))

macro "disj_intro_right" h:ident "from" p:term:arg hq:ident : tactic =>
  `(tactic|(have $h : ($p)∨_ := Or.inr $hq))

macro "impl_elim" hq:ident "from" hp:ident hpq:ident : tactic =>
  `(tactic|(have $hq := $hpq $hp))

namespace And
theorem elim' {P Q : Prop} {R : Sort} (h : P ∧ Q) (f : P → Q → R) : R := by
  sorry
end And

variable (S:Type) 
variable (P Q R : S → Prop)

example ( h : ( ∀ x : S, P x → Q x ) ∧ ( ∀ y : S, P y → R y ) ) : (∀ x : S, P x → Q x ∧ R x) := by
  sorry

/- The same proof using another idiom for conjunction elimination -/
example ( h : ( ∀ x : S, P x → Q x ) ∧ ( ∀ y : S, P y → R y ) ) : (∀ x : S, P x → Q x ∧ R x) := by
  sorry

/- The same proof using even another idiom for conjunction elimination -/
example ( h : ( ∀ x : S, P x → Q x ) ∧ ( ∀ y : S, P y → R y ) ) : (∀ x : S, P x → Q x ∧ R x) := by
  sorry

/- This example shows existential introduction, existential elimination, and universal elimination. -/
example ( h : ( ∃ x : S, P x ) ∧ ( ∀ x : S, Q x ) ) : ∃ x : S, P x ∧ Q x := by
  sorry

/- Here is another proof, using a different idiom for the existential elimination -/
example ( h : ( ∃ x : S, P x ) ∧ ( ∀ x : S, Q x ) ) : ∃ x : S, P x ∧ Q x := by
  sorry