let A be QC-alphabet ; 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)) is valid iff p => (All (x,q)) is valid )
let p, q be 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)) is valid iff p => (All (x,q)) is valid )
let x be bound_QC-variable of A; ( not x in still_not-bound_in p implies ( All (x,(p => q)) is valid iff p => (All (x,q)) is valid ) )
assume
not x in still_not-bound_in p
; ( All (x,(p => q)) is valid iff p => (All (x,q)) is valid )
then
(p => (All (x,q))) <=> (All (x,(p => q))) is valid
by Th75;
hence
( All (x,(p => q)) is valid iff p => (All (x,q)) is valid )
by Lm15; verum