let Al be QC-alphabet ; for A being non empty set
for v being Element of Valuations_in (Al,A)
for p, q being Element of CQC-WFF Al
for J being interpretation of Al,A holds
( J,v |= p => q iff ( J,v |= p implies J,v |= q ) )
let A be non empty set ; for v being Element of Valuations_in (Al,A)
for p, q being Element of CQC-WFF Al
for J being interpretation of Al,A holds
( J,v |= p => q iff ( J,v |= p implies J,v |= q ) )
let v be Element of Valuations_in (Al,A); for p, q being Element of CQC-WFF Al
for J being interpretation of Al,A holds
( J,v |= p => q iff ( J,v |= p implies J,v |= q ) )
let p, q be Element of CQC-WFF Al; for J being interpretation of Al,A holds
( J,v |= p => q iff ( J,v |= p implies J,v |= q ) )
let J be interpretation of Al,A; ( J,v |= p => q iff ( J,v |= p implies J,v |= q ) )
thus
( J,v |= p => q & J,v |= p implies J,v |= q )
by Th23; ( ( J,v |= p implies J,v |= q ) implies J,v |= p => q )
assume
( J,v |= p implies J,v |= q )
; J,v |= p => q
then
( (Valid (p,J)) . v = TRUE implies (Valid (q,J)) . v = TRUE )
;
then
( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE )
by XBOOLEAN:def 3;
hence
J,v |= p => q
by Th23; verum