Knowing that the equation for lift is:
L = CL ½ rho V2 S
And knowing that extending flaps will increase our wing surface area (S) and the coefficient lift because of the change in profile (CL), our lift will subsequently also be increased and therefore cause the aircraft to climb. This is of course assuming that there is no change in speed, air density, which can be assumed, but also as the question states, no change in pitch and therefore no change in angle of attack.
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