Proof:

Suppose we have a set of students. Different students study different subjects, with some of them studing several subjects at once. Suppose among many other subjects we have Biology and Chemistry. Then $P(B|C)$ would be probability of selecting a Biology student out of set of Chemistry students. 

Now we need to ask ourselves: what would be the complement of event "selecting a Biology student out of set of Chemistry students"?. Remember that we select a Biology student out of set of Chemistry students, NOT from all students. In other words, our sample space shrunk to the size of the set of Chemistry students. The complement event would be selecting a Chemistry student who does NOT study Biology (although the person can study other subjects along with Chemistry). 

The probability of selecting such person out of set of Chemistry students can be written as $P(\tilde B|C)$.

Thus $1-P(B|C)=P(\tilde B|C)$ because $P(B|C)$ and $P(\tilde B|C)$ are complements of each other.