The fact that D is the midpoint of arc ADC means that arcs AD and CD are of the same length. Because they lie under the exact same circle, it also follows that the two chords AD and CD are equal in length.

So, triangle ADC is isosceles, with AD = CD.

We can then see that EDC
= 180 - ADC
= 180 - (180 - ACD - DAC)
= ACD + DAC
= DAC + DAC (because of the isosceles triangle)
= 2 x DAC

Because ADC and ABC are opposite angles of the cyclic quadrilateral ABCD, these two angles (lying in opposite segments) are supplementary (add up to 180).

ABC + ADC = 180
ABC = 180 - ADC
ABC = EDC

From earlier, EDC = 2 x DAC, so...

ABC = 2 x DAC
ABD + CBD = 2 x DAC

But then, DAC = CBD because they are angles lying in the same segment. So...

ABD + CBD = 2 x CBD
ABD = 2 x CBD - CBD
ABD = CBD