let Al be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF Al
for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds
Ant f |= q
let p, q be Element of CQC-WFF Al; :: thesis: for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds
Ant f |= q
let f be FinSequence of CQC-WFF Al; :: thesis: ( Ant f |= p '&' q implies Ant f |= q )
assume A1: Ant f |= p '&' q ; :: thesis: Ant f |= q
let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds
J,v |= q
let J be interpretation of Al,A; :: thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds
J,v |= q
let v be Element of Valuations_in (Al,A); :: thesis: ( J,v |= Ant f implies J,v |= q )
assume J,v |= Ant f ; :: thesis: J,v |= q
then J,v |= p '&' q by A1;
hence J,v |= q by VALUAT_1:18; :: thesis: verum