The phrase "with that, someone does something" often denotes a causal relationship between events or actions. Some synonymous phrases for this include "consequently," "as a result," "therefore," "thus," and "hence." Alternatively, one could use phrases such as "because of this," "due to this," "accordingly," or "ergo" to convey similar meaning. This type of phrase can be especially useful in writing or speaking to connect thoughts or ideas and provide greater clarity. Ultimately, choosing the most appropriate synonym depends on context and the specific tone or style desired.