let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for x being bound_QC-variable of A holds (All (x,(p <=> q))) => ((All (x,p)) <=> (All (x,q))) is valid
let p, q be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A holds (All (x,(p <=> q))) => ((All (x,p)) <=> (All (x,q))) is valid
let x be bound_QC-variable of A; :: thesis: (All (x,(p <=> q))) => ((All (x,p)) <=> (All (x,q))) is valid
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 (All (x,(p <=> q))) => ((All (x,p)) <=> (All (x,q))) is valid by A2, LUKASI_1:42; :: thesis: verum