# Silver-like forcing preserves p-points (reference request)

A Silver forcing "below $$2^n$$" is defned e.g. in Definition 7.4.11 of [Tomek Bartoszyński and Haim Judah, Set Theory: on the structure of the real line, A. K. Peters, Wellesley, 1995.]. It is called Infinitely equal forcing EE there. In the same book, in Lemma 7.4.15 the authors show that EE preserves p-points. However, the proof of this lemma seems to be not complete/correct: the choice of conditions $$p^{n+1}$$ is not clear, as needed extensions of $$p^n$$ for various $$r_n^j$$ may "contradict" each other. (This would not be a problem if conditions were $$2^n$$-splitting trees, i.e., if the forcing were more like the Sacks rather than Silver.)

Do you know another source/reference for the proof that EE preserves p-points? Or perhaps I am missing something and the proof in the book is actually complete?