Reducing a system of two boolean algebra assertions to a single one
No.  There is no such binary operation.

This is easy to prove with a counting argument.  Given a candidate binary operation $?$, count the number $n$ of boolean values $x,y$ such that $x?y$ is true.   Out of the 16 possible values for $a,b,c,d$, exactly $m=n^2+(4-n)^2$ of them will satisfy $a?c = b?d$.  The possible values for $m$ are $m \in \{16,10,8\}$.  However, there are only 4 possible values of $a,b,c,d$ such that $a=b$ and $c=d$, which is not in the set of possible values for $m$.

Or, you could do as David Clarke suggests and write a tiny program to enumerate all possibilities.  Frankly, you probably should have done that before asking in any case; that would have helped you ask a more specific question, such as "I know no such binary operation exists; can you give me any insight why not?".