Example 4 If $\bar{B} \Rightarrow \bar{A}$ holds, and $\bar{A} \Rightarrow \bar{B}$ does not hold, then condition $B$ is ( ) for conclusion $A$ to hold.
(A) Sufficient but not necessary condition
(B) Necessary but not sufficient condition
(C) Sufficient and necessary condition
(D) Neither sufficient nor necessary condition