David's trick

We put into a general setting a technique of Rene' David (see "A Very Absolute Pi^1_2 Singleton, Annals of Pure and Applied Logic, 1982) to show that for S a Sigma^1_1 statement quantifying over subclasses of V of a special form, there is a stronger Sigma^1_3 statement quantifying over reals which can be forced over any model of S. Then I mention some application to Pi^1_2 Singletons and to Sigma^1_3 absoluteness.