Remove["`*"]

Vp  =  Vgp  +  (Vip - Vgp) Rfp /(Rip + Rfp)

Vm  =  
      Vout  +  (Vim - 
              Vout) Rfm /(Rim + Rfm)

Solve[Vout == A (Vp - Vm), Vout]

InputForm[%]

Vout = 
      Simplify[(A*
                Vgp - (A*Rfm*Vim)/(Rfm + Rim) + (A*
                    Rfp*(-Vgp + Vip))/(Rfp + Rip))/(1 + 
              A - (A*Rfm)/(Rfm + Rim))]

VoutNI  =  
      Limit[Vout  /.  {Vim -> 0, Rim -> Ri, Rfm -> Rf}, 
        Rfp -> Infinity]

(* 
      non - inverting amplifier with infinite open loop gain *) 
Limit[VoutNI, 
        A -> Infinity]  /.  (a_ + b_)/a_  ->  
        1 + b/a

VoutI  =  
      Limit[Vout  /.  {Rim -> Ri, Rfm -> Rf, Vip -> 0}, 
        Rfp -> Infinity]

(* 
      inverting amplifier with infinite open loop gain *) 
Limit[VoutI, A -> Infinity]

VoutVF  =  
      Limit[Limit[Vout, Rfp -> Infinity], 
        Rim -> Infinity]

(* 
      voltage follower with infinite open loop gain *) 
VoutVF  =  
      Limit[VoutVF, A -> Infinity]

Simplify[
      Vout  /. {Rfm -> Rf, Rfp -> Rf, Rim -> Ri, 
          Rip -> Ri, Vgp -> 0}]

Limit[
      Simplify[Vout  /. {Rfm -> Rf, Rfp -> Rf, Rim -> Ri, 
            Rip -> Ri, Vgp -> 0}], 
      A -> Infinity]