Phrase Study 2.to rest on/upon:(of a proof,argument,etc.) to depend on esp.be based on Your argument rests on a statement that can't be proved. We've tried everything;now it all rests on him. Phrase Study 2. to rest on/upon: (of a proof, argument, etc.) to depend on esp. be based on Your argument rests on a statement that can't be proved. We've tried everything; now it all rests on him