Simulating multiple-inheritance in Scala

I am working on a project that basically requires multiple-inheritance in the formal sense:

class X
class Y

class XY extends X,Y

I have two classes which have some definitions and are used throughout code, i.e:

def alpha(x: X)
def beta(y: Y)

I want to dynamically create a class XY which is just merges all definitions from X and Y but still retains the type-safety. More specifically that the previously defined alpha and beta still accept this merged class.

I know that Scala allows mixing in traits, for example:

trait T
class A

val ta = new A with T

That works fine.

However, I cannot do it with classes:

class X
class Y

val xy = new X with Y

As the with .. must be a trait.

I have tried circumventing this by doing the following:

trait xV extends X
trait yV extends Y

val xy = new xV with yV

Unfortunately this does not work and gives the following error:

Error:(33, 26) illegal inheritance; superclass X
 is not a subclass of the superclass Y
 of the mixin trait yV
    val xy = new xV with yV

Any help would be appreciated.

Edit:

To clarify, I cannot modify classes X or Y.

Solution

Doing this literally is impossible. But

More specifically that the previously defined alpha and beta still accept this merged class.

This specific requirement can be achieved by using implicit conversions:

class XY {
  val x = new X
  val y = new Y
}

object XY {
  implicit def toX(xy: XY): X = xy.x
  implicit def toY(xy: XY): Y = xy.y
}

You'll also be able to call X's and Y's methods directly.

But e.g. xy match { case x: X => ... won't match, and similarly xy.isInstanceOf[X] will be false. If X or Y override any of the Object methods: equals, hashCode, or toString, they won't be inherited by XY.