Shot down is a common phrase that implies rejection or disapproval. There are several synonyms that could replace this phrase depending on the context of the situation and intended tone. Here are some options: rejected, denied, dismissed, turned down, vetoed, opposed, rebuffed, discarded, refuted, contradicted, invalidated, snubbed, discredited, disavowed. Each of these synonyms has a slightly different nuance and implication. Some may be more appropriate for formal situations while others may be more suited for casual conversations. Choosing the right synonym can help convey a clear message and communicate effectively.