How to read the statement "If for every subset B' of B, it holds that α ∈ Cn(B') iff β ∈ Cn(B'), then B - α = B - β"?

Question

I'm confused specifically about the second part of the "iff".

For the first part "if for every subset B' of B, it holds that α ∈ Cn(B')" I understand that it is saying that α ∈ Cn(B') must hold for any subset B' of B.

But in the second part, specifically "if β ∈ Cn(B')", what B' is it refering to? Is it every subset B' as well or is it an arbitrary B'?

Answer 1

If for every subset B' of B, it holds that α ∈ C(B') iff β ∈ C(B'), then B - α = B - β

( for every subset B' of B, it holds that α ∈ C(B') iff β ∈ C(B') ) $\;\implies\;$ B - α = B - β

$\bigg(\forall B'{\subseteq}B\; \Big(α ∈ C(B') \iff β ∈ C(B')\Big) \bigg) \implies B - α = B - β$