Why is the electric field inside of a conducting sphere zero?

1. The problem statement, all variables and given/known data
Consider a solid conducting sphere with a radius a and charge Q1 on it. There is a conducting spherical shell concentric to the sphere. The shell has an inner radius b > a, outer radius c and a net charge Q2 on the shell. Denote the charge on the inner surface of the shell by Q′2 and that on the outer surface of the shell by Q′′2.

3. The attempt at a solution

I know the answer: you create a Gaussian surface within the outer conductor shell and somehow the electric field inside of it is zero. I’m trying to figure out why… is a charge supposed to collect on the inner surface that cancels out the charge on the inner sphere? How is that possible if the outer shell already has a charge on it? In order for the inner sphere to be canceled ( a charge of Q), the charge on the inner surface of the shell would have to be -Q, so does the negative charge basically induce a dipole on the outer shell?

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