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