(defun subeq (x y)
  (if (endp x)
      t
      (if (mem (first x) y)
          (subeq (rest x) y)
          nil)))

Theorem.
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (subeq a b) nil))

Proof.
My goal is to prove:
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (subeq a b) nil))
= {def subeq}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (if (endp a)
                    t
                    (if (mem (first a) b)
                        (subeq (rest a) b)
                        nil))
                nil))
= {hyp}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (if nil
                    t
                    (if (mem (first a) b)
                        (subeq (rest a) b)
                        nil))
                nil))
= {(if nil x y) = y}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (if (mem (first a) b)
                    (subeq (rest a) b)
                    nil)
                nil))
= {def subeq}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (if (mem (first a) b)
                    (if (endp (rest a))
                        t
                        (if (mem (first (rest a)) b)
                            (subeq (rest (rest a)) b)
                            nil))
                    nil)
                nil))
= {hyp}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (if (mem (first a) b)
                    (if nil
                        t
                        (if (mem (first (rest a)) b)
                            (subeq (rest (rest a)) b)
                            nil))
                    nil)
                nil))
= {(if nil x y) = y}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (if (mem (first a) b)
                    (if (mem (first (rest a)) b)
                        (subeq (rest (rest a)) b)
                        nil)
                    nil)
                nil))

= {hyp}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (if (mem (first a) b)
                    (if nil
                        (subeq (rest (rest a)) b)
                        nil)
                    nil)
                nil))
= {(if nil x y) = y}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal (if (mem (first a) b)
                    nil
                    nil)
                nil))
= {if-x-y-y: (if x y y) = y}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         (equal nil
                nil))
= {x=x}
(implies (and (not (endp a))
              (not (endp (rest a)))
              (not (mem (first (rest a)) b)))
         t)
= {(implies p t) = t}
t
Q.E.D.