reserve A,B,p,q,r,s for Element of LTLB_WFF,
  i,j,k,n for Element of NAT,
  X for Subset of LTLB_WFF,
  f,f1 for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN;

theorem Th54: X |- p => q & X |- p => r & X |- r => p implies X |- r => q
  proof
    assume that
A1: X |- p => q and
A2: X |- p => r and
A3: X |- r => p;
    (p => q) => (p => r => (r => p => (r => q))) is ctaut by Th46;
    then (p => q) => (p => r => (r => p => (r => q))) in LTL_axioms
    by LTLAXIO1:def 17;then
    X |- (p => q) => (p => r => (r => p => (r => q))) by LTLAXIO1:42;
    then X |- p => r => (r => p => (r => q)) by LTLAXIO1:43,A1;
    then X |- r => p => (r => q) by LTLAXIO1:43,A2;
    hence thesis by LTLAXIO1:43,A3;
  end;
