reserve X,Y for set, x,y,z for object, i,j,n for natural number;
reserve
  n for non empty Nat,
  S for non empty non void n PC-correct PCLangSignature,
  L for language MSAlgebra over S,
  F for PC-theory of L,
  A,B,C,D for Formula of L;

theorem
  (A\iffB)\imp(B\iffA) in F
  proof
A1: (B\impA)\and(A\impB)\imp(B\iffA) in F by Def38;
A2: (A\impB)\and(B\impA)\imp(B\impA)\and(A\impB) in F by Th50;
    (A\impB)\and(B\impA)\imp(B\impA)\and(A\impB)\imp
    (((B\impA)\and(A\impB)\imp(B\iffA))\imp
    ((A\impB)\and(B\impA)\imp(B\iffA))) in F by Th39; then
    (((B\impA)\and(A\impB)\imp(B\iffA))\imp
    ((A\impB)\and(B\impA)\imp(B\iffA))) in F by A2,Def38; then
A3: (A\impB)\and(B\impA)\imp(B\iffA) in F by A1,Def38;
A4: (A\iffB)\imp(A\impB)\imp(((A\iffB)\imp(B\impA))\imp
    ((A\iffB)\imp(A\impB)\and(B\impA))) in F by Th49;
A5: (A\iffB)\imp(A\impB) in F & ((A\iffB)\imp(B\impA)) in F by Def38; then
    (((A\iffB)\imp(B\impA))\imp((A\iffB)\imp(A\impB)\and(B\impA))) in F
    by A4,Def38; then
A6: (A\iffB)\imp(A\impB)\and(B\impA) in F by A5,Def38;
    (A\iffB)\imp(A\impB)\and(B\impA)\imp(((A\impB)\and(B\impA)\imp(B\iffA))\imp
    ((A\iffB)\imp(B\iffA))) in F by Th39; then
    ((A\impB)\and(B\impA)\imp(B\iffA))\imp((A\iffB)\imp(B\iffA)) in F
    by A6,Def38;
    hence thesis by A3,Def38;
  end;
