let Al be QC-alphabet ; 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 ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) )
let A be 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 ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) )
let v be 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 ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) )
let p, q be Element of CQC-WFF Al; for J being interpretation of Al,A holds
( J,v |= p => q iff ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) )
let J be interpretation of Al,A; ( J,v |= p => q iff ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) )
A1:
now ( ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) implies J,v |= p => q )A2:
now ( (Valid (q,J)) . v = TRUE implies J,v |= p => q )assume A3:
(Valid (q,J)) . v = TRUE
;
J,v |= p => qassume
not
J,
v |= p => q
;
contradictionthen
(Valid ((p => q),J)) . v <> TRUE
;
then
(Valid ((p => q),J)) . v = FALSE
by XBOOLEAN:def 3;
then
(Valid (('not' (p '&' ('not' q))),J)) . v = FALSE
by QC_LANG2:def 2;
then
'not' ((Valid ((p '&' ('not' q)),J)) . v) = FALSE
by Th10;
then
(Valid ((p '&' ('not' q)),J)) . v = TRUE
by MARGREL1:11;
then
((Valid (p,J)) . v) '&' ((Valid (('not' q),J)) . v) = TRUE
by Th12;
then
((Valid (p,J)) . v) '&' ('not' ((Valid (q,J)) . v)) = TRUE
by Th10;
then
'not' ((Valid (q,J)) . v) = TRUE
by MARGREL1:12;
hence
contradiction
by A3, MARGREL1:11;
verum end; assume
(
(Valid (p,J)) . v = FALSE or
(Valid (q,J)) . v = TRUE )
;
J,v |= p => qhence
J,
v |= p => q
by A4, A2;
verum end;
now ( not J,v |= p => q or (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE )assume
J,
v |= p => q
;
( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE )then
(Valid ((p => q),J)) . v = TRUE
;
then
(Valid (('not' (p '&' ('not' q))),J)) . v = TRUE
by QC_LANG2:def 2;
then
'not' ((Valid ((p '&' ('not' q)),J)) . v) = TRUE
by Th10;
then
(Valid ((p '&' ('not' q)),J)) . v = FALSE
by MARGREL1:11;
then
((Valid (p,J)) . v) '&' ((Valid (('not' q),J)) . v) = FALSE
by Th12;
then
((Valid (p,J)) . v) '&' ('not' ((Valid (q,J)) . v)) = FALSE
by Th10;
then
(
(Valid (p,J)) . v = FALSE or
'not' ((Valid (q,J)) . v) = FALSE )
by MARGREL1:12;
hence
(
(Valid (p,J)) . v = FALSE or
(Valid (q,J)) . v = TRUE )
by MARGREL1:11;
verum end;
hence
( J,v |= p => q iff ( (Valid (p,J)) . v = FALSE or (Valid (q,J)) . v = TRUE ) )
by A1; verum