let Al be QC-alphabet ; :: thesis: 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 ; :: thesis: 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); :: thesis: 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; :: thesis: 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; :: thesis: ( J,v |= p => q iff ( J,v |= p implies J,v |= q ) )

hereby :: thesis: ( ( J,v |= p implies J,v |= q ) implies J,v |= p => q )

assume J,v |= p => q ; :: thesis: ( J,v |= p implies J,v |= q )
then ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) by Th23;
hence ( J,v |= p implies J,v |= q ) by Def7; :: thesis: verum

end;

assume ( J,v |= p implies J,v |= q ) ; :: thesis: J,v |= p => q
then ( (Valid (p,J)) . v = TRUE implies (Valid (q,J)) . v = TRUE ) by Def7;
then ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) by XBOOLEAN:def 3;
hence J,v |= p => q by Th23; :: thesis: verum