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 Th37:
  (B\impC)\imp((A\impB)\imp(A\impC)) in F
proof
A1: ((A\imp(B\impC))\imp((A\impB)\imp(A\impC))) in F by Def38;
A2: ( (B\impC)\imp((A\imp(B\impC))\imp((A\impB)\imp(A\impC))))\imp(((B\impC)
    \imp(A\imp(B\impC)))\imp((B\impC)\imp((A\impB)\imp(A\impC)))) in F
    by Def38;
    ((A\imp(B\impC))\imp((A\impB)\imp(A\impC)))\imp( (B\impC)\imp((A\imp(B
    \impC))\imp((A\impB)\imp(A\impC)))) in F by Def38;
    then
    ( (B\impC)\imp((A\imp(B\impC))\imp((A\impB)\imp(A\impC)))) in F
    by A1,Def38;
    then
A3: ((B\impC)\imp(A\imp(B\impC)))\imp((B\impC)\imp((A\impB)\imp(A\impC))) in F
    by A2,Def38;
    (B\impC)\imp(A\imp(B\impC)) in F by Def38;
    hence thesis by A3,Def38;
  end;
