let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for x being bound_QC-variable of A st not x in still_not-bound_in p holds
( (All (x,(p => q))) => (p => (All (x,q))) is valid & (p => (All (x,q))) => (All (x,(p => q))) is valid )
let p, q be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A st not x in still_not-bound_in p holds
( (All (x,(p => q))) => (p => (All (x,q))) is valid & (p => (All (x,q))) => (All (x,(p => q))) is valid )
let x be bound_QC-variable of A; :: thesis: ( not x in still_not-bound_in p implies ( (All (x,(p => q))) => (p => (All (x,q))) is valid & (p => (All (x,q))) => (All (x,(p => q))) is valid ) )
assume A1: not x in still_not-bound_in p ; :: thesis: ( (All (x,(p => q))) => (p => (All (x,q))) is valid & (p => (All (x,q))) => (All (x,(p => q))) is valid )
hence (All (x,(p => q))) => (p => (All (x,q))) is valid by Lm17; :: thesis: (p => (All (x,q))) => (All (x,(p => q))) is valid
not x in still_not-bound_in (All (x,q)) by Th5;
then not x in still_not-bound_in (p => (All (x,q))) by A1, Th7;
then A2: (All (x,((p => (All (x,q))) => (p => q)))) => ((p => (All (x,q))) => (All (x,(p => q)))) is valid by Lm17;
( All (x,(((All (x,q)) => q) => ((p => (All (x,q))) => (p => q)))) is valid & (All (x,(((All (x,q)) => q) => ((p => (All (x,q))) => (p => q))))) => ((All (x,((All (x,q)) => q))) => (All (x,((p => (All (x,q))) => (p => q))))) is valid ) by Th23, Th30;
then A3: (All (x,((All (x,q)) => q))) => (All (x,((p => (All (x,q))) => (p => q)))) is valid by CQC_THE1:65;
All (x,((All (x,q)) => q)) is valid by Th23, CQC_THE1:66;
then All (x,((p => (All (x,q))) => (p => q))) is valid by A3, CQC_THE1:65;
hence (p => (All (x,q))) => (All (x,(p => q))) is valid by A2, CQC_THE1:65; :: thesis: verum