fluid dynamics - Why can Bernoulli's equation be applied along two different streamlines for an irrotational flow?

The following is written in Physics by Halliday, Resnick and Krane (5th ed.).

$$p + \frac{1}{2} \rho v^2 + \rho g y = \mathrm{constant}$$

Strictly speaking, the points to which we apply Bernoulli's equation should be along the same streamline. However, if the flow is irrotational, the value of the constant is same for all the streamlines in the tube of flow, so Bernoulli's equation can be applied to any two points in the flow.

Here, $p$ is the pressure of the fluid at a point, $\rho$ is the density(assumed constant), $v$ is the velocity of the fluid element, and $y$ is the vertical distance of the element from a fixed reference point. From point to point, $p$, $v$ and $y$ will change.

How can one prove that the constant in Bernoulli's equation does not change along two streamlines for irrotational flow?

The fluid can be assumed to be non-viscous, incompressible and the flow is steady.