T = true
F = false
tval(x) = truth value of x
LHS = Left hand side of the equivalence
RHS = Right hand side of the equivalence
Universal quantifier is not distributive over disjunction. Therefore ∀xP(x)∨∀xQ(x)≡∀x(P(x)∨Q(x)) is false.
Let Domain = ℝ
P(x) = x is rational
Q(x) = x is irrational
tval(LHS) = F but tval(RHS) =T
The statements "All real numbers are rational." and "All real numbers are irrational" are both false. That makes the left-hand side of the equivalence false as well.
However, if you say all real numbers are either rational or irrational, that makes the statement true because that is the very definition of real numbers.