Are these two integrals equal?

Calculus Level pending

Determine if these statements are true:

Statement A: These two integrals are equal:
\[ I_{1}=\int_{0}^{1} x^x \; dx, \qquad I_{2}=\int_{0}^{1} \int_{0}^{1} (xy)^{xy} \; dy \; d x. \]

Statement B: These two integrals are equal: \[ I_{1}=\int_{0}^{1} x^x \; dx, \qquad I_{3}=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} (xyz)^{xyz} \; \mathrm{d} z \; \mathrm{d}y \; dx. \]

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