reserve A for QC-alphabet;
reserve X,T for Subset of CQC-WFF(A);
reserve F,G,H,p,q,r,t for Element of CQC-WFF(A);
reserve s,h for QC-formula of A;
reserve x,y for bound_QC-variable of A;
reserve f for FinSequence of [:CQC-WFF(A),Proof_Step_Kinds:];
reserve i,j for Element of NAT;

theorem Th32:
  All(x,p <=> q) => (All(x,p) <=> All(x,q)) is valid
proof
A1: All(x,(p => q) '&' (q => p)) => ((p => q) '&' (q => p)) is valid by
CQC_THE1:66;
  (p <=> q) => (p <=> q) is valid;
  then (p <=> q) => ((p => q) '&' (q => p)) is valid by QC_LANG2:def 4;
  then All(x,(p <=> q) => ((p => q) '&' (q => p))) is valid by Th23;
  then
A2: All(x,p <=> q) => All(x,(p => q) '&' (q => p)) is valid by Th31;
  All(x,p => q) => (All(x,p) => All(x,q)) is valid & All(x,q => p) => (
  All(x,q ) => All(x,p)) is valid by Th30;
  then
  (All(x,p => q) '&' All(x,q => p)) => ((All(x,p) => All(x,q)) '&' (All(x
  ,q) => All(x,p))) is valid by Lm5;
  then
A3: (All(x,p => q) '&' All(x,q => p)) => (All(x,p) <=> All(x,q)) is valid
  by QC_LANG2:def 4;
A4: not x in still_not-bound_in All(x,(p => q) '&' (q => p)) by Th5;
  ((p => q) '&' (q => p)) => (q => p) is valid by Lm1;
  then All(x,(p => q) '&' (q => p)) => (q => p) is valid by A1,LUKASI_1:42;
  then
A5: All(x,(p => q) '&' (q => p)) => All(x,q => p) is valid by A4,CQC_THE1:67;
  ((p => q) '&' (q => p)) => (p => q) is valid by Lm1;
  then All(x,(p => q) '&' (q => p)) => (p => q) is valid by A1,LUKASI_1:42;
  then All(x,(p => q) '&' (q => p)) => All(x,p => q) is valid by A4,CQC_THE1:67
;
  then All(x,(p => q) '&' (q => p)) => (All(x,p => q) '&' All(x,q => p)) is
  valid by A5,Lm3;
  then All(x,(p => q) '&' (q => p)) => (All(x,p) <=> All(x,q)) is valid by A3,
LUKASI_1:42;
  hence thesis by A2,LUKASI_1:42;
end;
