Antonyms for the phrase "be consequent on" are "not related," "unrelated," "unconnected," "independent," "random," and "arbitrary." When an event is said to be consequent on something, it implies that it was caused by or is a result of that thing. However, the aforementioned antonyms indicate that the event and the cause are not connected in any way. Saying that something is unrelated, independent or arbitrary means that it has nothing to do with the cause. On the other hand, describing an event as random means that it happened by chance, without any apparent cause or reason.