# Unclear manipulation with forcing conditions

To this question in Kunen's introduction to independence proofs

on the page 379, in the proof of lemma V.7.3 you never mention $$q$$ in the proof, only $$p$$; why? I cannot make sense of this simple proof and lemma.

The claim is that there is a q extending p and a countable subset B of kappa such that q forces a certain property of B.

The proof says that if there is no such q and B, then p forces that in the generic extension, there is no countable ground-model subset B of kappa satisfying this property.

And this last statement (which doesn't mention q) leads to a contradiction.

BUT I still do not understand why we can get rid off q and say

then p forces that in the generic extension, there is no countable ground-model subset B of kappa satisfying this property.

I may write down the details for those who do not have the book opened on the table, but I think that the context is almost clear from the question and the partial answer of it.

The inductive definition of forcing (by complexity of formulas) gives in particular that $$p$$ forces $$\lnot\psi$$ if and only if no extension of $$p$$ forces $$\psi$$. That is exactly what is being claimed.