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
( (All x,p) => q is valid iff Ex x,(p => q) is valid )
let x be bound_QC-variable; :: thesis: ( not x in still_not-bound_in q implies ( (All x,p) => q is valid iff Ex x,(p => q) is valid ) )
assume not x in still_not-bound_in q ; :: thesis: ( (All x,p) => q is valid iff Ex x,(p => q) is valid )
then A1: (Ex x,(p => q)) => ((All x,p) => q) is valid by Th81;
((All x,p) => q) => (Ex x,(p => q)) is valid by Th82;
then ((All x,p) => q) <=> (Ex x,(p => q)) is valid by A1, Lm14;
hence ( (All x,p) => q is valid iff Ex x,(p => q) is valid ) by Lm15; :: thesis: verum