let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable 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; :: 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
A2: All x,((All x,q) => q) is valid by Th26, CQC_THE1:105;
A3: All x,(((All x,q) => q) => ((p => (All x,q)) => (p => q))) is valid by Th26, LUKASI_1:59;
(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 Th34;
then (All x,((All x,q) => q)) => (All x,((p => (All x,q)) => (p => q))) is valid by A3, CQC_THE1:104;
then A4: All x,((p => (All x,q)) => (p => q)) is valid by A2, CQC_THE1:104;
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 (All x,((p => (All x,q)) => (p => q))) => ((p => (All x,q)) => (All x,(p => q))) is valid by Lm17;
hence (p => (All x,q)) => (All x,(p => q)) is valid by A4, CQC_THE1:104; :: thesis: verum