Simplifying Sums of Product Expression obtained from 8-3 Priority Encoder (Computer Science)

Question

I have an example for simplifying expressions in sums of product form, but I can't figure out which algebraic theorem was used to get rid of some of the variables:

Step 1. (A'B'C'D'E'F'G) + (A'B'C'D'E) + (A'B'C) + (A)

which simplifies to:

Step 2. (B'D'F'G + B'D'E + B'C + A)

Which simplifies to:

Step 3. [B'(D'F'G + D'E + C) + A]

At step 2, I am confused at how the A', C', and E' terms are gone. Can anyone explain which algebraic properties were used, or show me how the simplification was done?

Thanks!

This example was obtained from here

Answer 1

The identity

x + x'y = x + y 

is repeated used above for simplification

For example.,

A'B'C'D'E + A'B'C'D'E'F'G = A'B'C'D'[E + E'(F'G)] = A'B'C'D'[E + F'G] 
A'B'C + A'B'C'D'E = A'B'[C + C'D'E] = A'B'[C + C'(D'E)] = A'B'[C + D'E]