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 q holds
( (All (x,p)) => q is valid iff Ex (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 q holds
( (All (x,p)) => q is valid iff Ex (x,(p => q)) is valid )
let x be bound_QC-variable of A; :: 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 Th77;
((All (x,p)) => q) => (Ex (x,(p => q))) is valid by Th78;
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