How do I write Ax ( P(x) and Q(x) ) in Coq?

Question

I'm trying out Coq, but I'm not completely sure what I'm doing. Is:

Theorem new_theorem : forall x, P:Prop /\ Q:Prop

Equivalent to:

Ax ( P(x) and Q(x) )

(where A is supposed to be the universal quantifier).

Edit: I think they are.

Answer 1

Are you having problems with the syntax?

$ coqtop
Welcome to Coq 8.1pl3 (Dec. 2007)

Coq < Section Test.

Coq < Variable X:Set.
X is assumed

Coq < Variables P Q:X -> Prop.
P is assumed
Q is assumed

Coq < Theorem forall_test: forall x:X, P(x) /\ Q(x).
1 subgoal

  X : Set
  P : X -> Prop
  Q : X -> Prop
  ============================
   forall x : X, P x /\ Q x

forall_test <

Answer 2

Well, to answer your question:

Section test.

  Variable A : Type.           (* assume some universe A *)
  Variable P Q : A -> Prop.    (* and two predicates over A, P and Q *)

  Goal forall x, P x /\ Q x.   (* Ax, ( P(x) and Q(x) ) *)

End test.