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(1) $\forall x P(x) \wedge \forall x Q(x)$

(2) $\forall x (P(x) \wedge Q(x)) $

(3) $\forall y (\forall x P(x) \wedge Q(y))$

(4) $ \forall y \forall x(P(x)\wedge Q(y))$

I'm sure that (1) and (2) are equivalance. I think (3) and (4) are the same as (1) and (2) as $$\forall x (B \wedge P(x)) \Leftrightarrow B \wedge \forall x(P(x)),$$ but was not sure.

Anyone have ideas about this?


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up vote 2 down vote accepted

Yes, all four statements are equivalent. They all express that every element of the universe satisfies both $P$ and $Q$.

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