reserve a,b,c for boolean object;
reserve p,q,r,s,A,B,C for Element of LTLB_WFF,
        F,G,X,Y for Subset of LTLB_WFF,
        i,j,k,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;
reserve M for LTLModel;

theorem Th52:
  X|-p=>q implies X|-('not' q)=>('not' p)
 proof
  set A=(p=>q)=>(('not' q)=>('not' p));
  now let f be Function of LTLB_WFF,BOOLEAN;
   A1: (VAL f).p=0 or(VAL f).p=1 by XBOOLEAN:def 3;
   A2: (VAL f).q=0 or(VAL f).q=1 by XBOOLEAN:def 3;
   thus(VAL f).A=(VAL f).(p=>q)=>(VAL f).(('not' q)=>('not' p)) by Def15
    .=((VAL f).p=>(VAL f).q)=>(VAL f).(('not' q)=>('not' p)) by Def15
    .=((VAL f).p=>(VAL f).q)=>((VAL f).(q=>TFALSUM)=>(VAL f).(p=>TFALSUM)) by
Def15
    .=((VAL f).p=>(VAL f).q)=>(((VAL f).q=>(VAL f).TFALSUM)=>((VAL f).(p=>
TFALSUM))) by Def15
    .=((VAL f).p=>(VAL f).q)=>(((VAL f).q=>FALSE)=>((VAL f).(p=>TFALSUM))) by
Def15
    .=((VAL f).p=>(VAL f).q)=>(((VAL f).q=>FALSE)=>(((VAL f).p=>(VAL f).TFALSUM
))) by Def15
    .=1 by A1,A2;
  end;
  then A is LTL_TAUT_OF_PL;
  then A in LTL_axioms by Def17;
  then A3: X|-A by Th42;
  assume X|-p=>q;
  hence thesis by A3,Th43;
 end;
