let A be non empty set ; :: thesis: for v being Element of Valuations_in A
for p, q, t being Element of CQC-WFF
for J being interpretation of A holds J,v |= (p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))
let v be Element of Valuations_in A; :: thesis: for p, q, t being Element of CQC-WFF
for J being interpretation of A holds J,v |= (p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))
let p, q, t be Element of CQC-WFF ; :: thesis: for J being interpretation of A holds J,v |= (p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))
let J be interpretation of A; :: thesis: J,v |= (p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))
( p => q = 'not' (p '&' ('not' q)) & ('not' (q '&' t)) => ('not' (p '&' t)) = 'not' (('not' (q '&' t)) '&' ('not' ('not' (p '&' t)))) ) by QC_LANG2:def 2;
then A1: (Valid (((p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))),J)) . v = (Valid (('not' (('not' (p '&' ('not' q))) '&' ('not' ('not' (('not' (q '&' t)) '&' ('not' ('not' (p '&' t)))))))),J)) . v by QC_LANG2:def 2
.= 'not' ((Valid ((('not' (p '&' ('not' q))) '&' ('not' ('not' (('not' (q '&' t)) '&' ('not' ('not' (p '&' t))))))),J)) . v) by Th20
.= 'not' (((Valid (('not' (p '&' ('not' q))),J)) . v) '&' ((Valid (('not' ('not' (('not' (q '&' t)) '&' ('not' ('not' (p '&' t)))))),J)) . v)) by Th22 ;
A2: (Valid (('not' (p '&' ('not' q))),J)) . v = 'not' ((Valid ((p '&' ('not' q)),J)) . v) by Th20
.= 'not' (((Valid (p,J)) . v) '&' ((Valid (('not' q),J)) . v)) by Th22
.= 'not' (((Valid (p,J)) . v) '&' ('not' ((Valid (q,J)) . v))) by Th20 ;
(Valid (('not' ('not' (('not' (q '&' t)) '&' ('not' ('not' (p '&' t)))))),J)) . v = 'not' ((Valid (('not' (('not' (q '&' t)) '&' ('not' ('not' (p '&' t))))),J)) . v) by Th20
.= 'not' ('not' ((Valid ((('not' (q '&' t)) '&' ('not' ('not' (p '&' t)))),J)) . v)) by Th20
.= ((Valid (('not' (q '&' t)),J)) . v) '&' ((Valid (('not' ('not' (p '&' t))),J)) . v) by Th22
.= ('not' ((Valid ((q '&' t),J)) . v)) '&' ((Valid (('not' ('not' (p '&' t))),J)) . v) by Th20
.= ('not' ((Valid ((q '&' t),J)) . v)) '&' ('not' ((Valid (('not' (p '&' t)),J)) . v)) by Th20
.= ('not' ((Valid ((q '&' t),J)) . v)) '&' ('not' ('not' ((Valid ((p '&' t),J)) . v))) by Th20
.= ('not' (((Valid (q,J)) . v) '&' ((Valid (t,J)) . v))) '&' ((Valid ((p '&' t),J)) . v) by Th22
.= ('not' (((Valid (q,J)) . v) '&' ((Valid (t,J)) . v))) '&' (((Valid (p,J)) . v) '&' ((Valid (t,J)) . v)) by Th22 ;
hence (Valid (((p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))),J)) . v = TRUE by A1, A2, Lm2; :: according to VALUAT_1:def 7 :: thesis: verum