let q, p be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable st not x in still_not-bound_in q holds
( ((Ex x,p) => q) => (All x,(p => q)) is valid & (All x,(p => q)) => ((Ex x,p) => q) is valid )
let x be bound_QC-variable; :: thesis: ( not x in still_not-bound_in q implies ( ((Ex x,p) => q) => (All x,(p => q)) is valid & (All x,(p => q)) => ((Ex x,p) => q) is valid ) )
assume A1: not x in still_not-bound_in q ; :: thesis: ( ((Ex x,p) => q) => (All x,(p => q)) is valid & (All x,(p => q)) => ((Ex x,p) => q) is valid )
p => (Ex x,p) is valid by Th18;
then A2: ((Ex x,p) => q) => (p => q) is valid by LUKASI_1:42;
not x in still_not-bound_in (Ex x,p) by Th6;
then not x in still_not-bound_in ((Ex x,p) => q) by A1, Th7;
hence ((Ex x,p) => q) => (All x,(p => q)) is valid by A2, CQC_THE1:106; :: thesis: (All x,(p => q)) => ((Ex x,p) => q) is valid
(All x,(p => q)) => ((Ex x,p) => (Ex x,q)) is valid by Th38;
then A3: ((All x,(p => q)) '&' (Ex x,p)) => (Ex x,q) is valid by Th1;
(Ex x,q) => q is valid by A1, Th23;
then ((All x,(p => q)) '&' (Ex x,p)) => q is valid by A3, LUKASI_1:43;
hence (All x,(p => q)) => ((Ex x,p) => q) is valid by Th3; :: thesis: verum