Show that ∀xP(x)∧∃xQ(x) is logically equivalent to ∀x∃y(P(x)∧Q(y)), where all quantifiers have the same nonempty domain.
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We will start from the first given statement,
∀xP(x)∧∃xQ(x)
It doesn't matter if the variables are called x or called y, thus let us recall the variable in the second expression of the conjunction:
∀xP(x)∧∃yQ(y)
(∀xP(x))∧(∃yQ(y))
∀x(P(x)∧(∃yQ(y)))
Using the commutative law,
∀x((∃yQ(y))∧P(x))
∀x∃y(Q(y)∧P(x))
Again using the commutative law,
∀x∃y(P(x)∧Q(y))
Hence we can say that the statement ∀xP(x)∧∃xQ(x) is logically equivalent to ∀x∃y(P(x)∧Q(y)) where all quantifiers have the same nonempty domain.