reserve A for QC-alphabet;
reserve p, q, r, s, p1, q1 for Element of CQC-WFF(A),
  X, Y, Z, X1, X2 for Subset of CQC-WFF(A),
  h for QC-formula of A,
  x, y for bound_QC-variable of A,
  n for Element of NAT;

theorem
  p <==> q implies p |-| q
proof
  assume p <==> q;
  then p <=> q is valid;
  then (p => q) '&' (q => p) is valid by QC_LANG2:def 4;
  then
A1: {}(CQC-WFF(A)) |- (p => q) '&' (q => p) by CQC_THE1:def 9;
  then {}(CQC-WFF(A)) |- q => p by Lm3;
  then
A2: q => p is valid by CQC_THE1:def 9;
  {}(CQC-WFF(A)) |- p => q by A1,Lm3;
  then p => q is valid by CQC_THE1:def 9;
  hence p |- q & q |- p by A2,Th39;
end;
