let Al be QC-alphabet ; for A being non empty set
for v being Element of Valuations_in (Al,A)
for p, q, t being Element of CQC-WFF Al
for J being interpretation of Al,A holds J,v |= (p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))
let A be non empty set ; for v being Element of Valuations_in (Al,A)
for p, q, t being Element of CQC-WFF Al
for J being interpretation of Al,A holds J,v |= (p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))
let v be Element of Valuations_in (Al,A); for p, q, t being Element of CQC-WFF Al
for J being interpretation of Al,A holds J,v |= (p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))
let p, q, t be Element of CQC-WFF Al; for J being interpretation of Al,A holds J,v |= (p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))
let J be interpretation of Al,A; 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 Th10
.=
'not' (((Valid (('not' (p '&' ('not' q))),J)) . v) '&' ((Valid (('not' ('not' (('not' (q '&' t)) '&' ('not' ('not' (p '&' t)))))),J)) . v))
by Th12
;
A2: (Valid (('not' (p '&' ('not' q))),J)) . v =
'not' ((Valid ((p '&' ('not' q)),J)) . v)
by Th10
.=
'not' (((Valid (p,J)) . v) '&' ((Valid (('not' q),J)) . v))
by Th12
.=
'not' (((Valid (p,J)) . v) '&' ('not' ((Valid (q,J)) . v)))
by Th10
;
(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 Th10
.=
'not' ('not' ((Valid ((('not' (q '&' t)) '&' ('not' ('not' (p '&' t)))),J)) . v))
by Th10
.=
((Valid (('not' (q '&' t)),J)) . v) '&' ((Valid (('not' ('not' (p '&' t))),J)) . v)
by Th12
.=
('not' ((Valid ((q '&' t),J)) . v)) '&' ((Valid (('not' ('not' (p '&' t))),J)) . v)
by Th10
.=
('not' ((Valid ((q '&' t),J)) . v)) '&' ('not' ((Valid (('not' (p '&' t)),J)) . v))
by Th10
.=
('not' ((Valid ((q '&' t),J)) . v)) '&' ('not' ('not' ((Valid ((p '&' t),J)) . v)))
by Th10
.=
('not' (((Valid (q,J)) . v) '&' ((Valid (t,J)) . v))) '&' ((Valid ((p '&' t),J)) . v)
by Th12
.=
('not' (((Valid (q,J)) . v) '&' ((Valid (t,J)) . v))) '&' (((Valid (p,J)) . v) '&' ((Valid (t,J)) . v))
by Th12
;
hence
(Valid (((p => q) => (('not' (q '&' t)) => ('not' (p '&' t)))),J)) . v = TRUE
by A1, A2, Lm2; VALUAT_1:def 7 verum