(( A implies B ) v ( A implies D )) implies ( B v D )

(( A => B ) v ( A => D )) => ( B v D )

1. change the implications in the minor expressions into OR

(( not A v B) v (not A v D)) => (B v D)

2. reduce common variables in expressions

(not A v (B v D)) => (B v D)

3. change the implication in the major expression into OR

not (not A v (B v D)) v (B v D)

4. if we have a negation outside of brackets, then we can bring inside using de morgans laws.

(A ^ not (B v D)) v (B v D)

5. using the distributive laws, this will become.

(A v (B v D)) ^ ( not (B v D) v (B v D) )

6. get rid of the tautology (not (B v D) v (B v D))

(A v (B v D)) ^ 1

This is the same as:

A v (B v D)

7. now get rid of the brackets we dont need and we end up with:

A v B v D