Mutatis mutandis is Latin for what needs changing having been changed.
It can be used in place of words like analogously and similarly, and expresses that an argument can be applied to a new situation in an intuitive fashion.
In this case, this comes down to replacing $\cup$ by $\cap$, and $\lor$ by $\land$.
In particular, the essential flow of the argument was not changed in adapting the proof.
Also defined as
- The term is often used when some work but no imagination is demanded of the reader.