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 Th58:
  p <==> q implies All(x,p) <==> All(x,q)
proof
  assume
A1: p <==> q;
  then q => p is valid by Th50;
  then All(x,q => p) is valid by CQC_THE2:23;
  then
A2: All(x,q) => All(x,p) is valid by CQC_THE2:31;
  p => q is valid by A1,Th50;
  then All(x,p => q) is valid by CQC_THE2:23;
  then All(x,p) => All(x,q) is valid by CQC_THE2:31;
  hence thesis by A2,Th50;
end;
