She is his sister.
Bill's brother's wife's mother in law = Bill's mother
Bill's mother's daughter = Bill's father's daughter = Bill's sister
Proved.

She is Bill's sister.
Bill's brother's wife's mother in law = Bill's mother
Bill's mother's daughter = Bill's father's daughter = Bill's sister
Proved.

She is his sister.

Bill's brother's wife's mother in law = Bill's mother

Bill's mother's daughter = Bill's father's daughter = Bill's sister

Proved.

She is his sister.
Bill's brother's wife's mother in law = Bill's mother
Bill's mother's daughter = Bill's father's daughter = Bill's sister
Proved.