Explanation:

To determine which first-order logic formula is valid, let's analyze each option:

Option A: x(P(x) = Q(x)) = (xP(x) = xQ(x))
This option states that for any x, the predicate P(x) is equivalent to Q(x).
To evaluate the validity of this formula, we can consider an example where P(x) is "x is even" and Q(x) is "x is divisible by 2". In this case, the formula holds true because the evenness of a number is equivalent to its divisibility by 2. Therefore, option A is valid.

Option B: x(P(x) ∧ Q(x)) = (xP(x) = xQ(x))
This option states that for any x, the conjunction of P(x) and Q(x) is equivalent to the equality of xP(x) and xQ(x).
To evaluate the validity of this formula, we can consider an example where P(x) is "x is prime" and Q(x) is "x is odd". In this case, the formula does not hold true because the conjunction of being prime and odd is not equivalent to the equality of the product of x and being prime with the product of x and being odd. Therefore, option B is not valid.

Option C: x(P(x) ∧ Q(x)) = (xP(x) ∧ xQ(x))
This option states that for any x, the conjunction of P(x) and Q(x) is equivalent to the conjunction of xP(x) and xQ(x).
To evaluate the validity of this formula, we can consider an example where P(x) is "x is even" and Q(x) is "x is divisible by 2". In this case, the formula does not hold true because the conjunction of being even and divisible by 2 is not equivalent to the conjunction of x being even and x being divisible by 2. Therefore, option C is not valid.

Option D: xy P(x, y) = yx P(x, y)
This option states that for any x and y, the predicate P(x, y) is equivalent regardless of the order of x and y.
To evaluate the validity of this formula, we can consider an example where P(x, y) is "x is the parent of y". In this case, the formula does not hold true because the parent-child relationship is not commutative. Therefore, option D is not valid.

Conclusion:
Based on the analysis, option A is the only valid first-order logic formula.