Biba's trick, with applications

We give another bit of evidence that forcing axioms provide proper framework for rigidity of quotient structures, by improving the OCA lifting theorem proved by the author in late 20th century and greatly simplifying its proof. In the assumptions of this theorem. We also extend the conclusion of author's 2004 lifting theorem from a lifting result for countably 3204-determined ideals to one for countably 80-determined ideals and weaken its assumptions.