let A be non empty set ; :: thesis: for x being bound_QC-variable
for v being Element of Valuations_in A
for p being Element of CQC-WFF
for J being interpretation of A holds J,v |= (All x,p) => p
let x be bound_QC-variable; :: thesis: for v being Element of Valuations_in A
for p being Element of CQC-WFF
for J being interpretation of A holds J,v |= (All x,p) => p
let v be Element of Valuations_in A; :: thesis: for p being Element of CQC-WFF
for J being interpretation of A holds J,v |= (All x,p) => p
let p be Element of CQC-WFF ; :: thesis: for J being interpretation of A holds J,v |= (All x,p) => p
let J be interpretation of A; :: thesis: J,v |= (All x,p) => p
thus
(Valid ((All x,p) => p),J) . v = TRUE
:: according to VALUAT_1:def 12 :: thesis: verumproof
assume
not
(Valid ((All x,p) => p),J) . v = TRUE
;
:: thesis: contradiction
then A1:
(Valid ((All x,p) => p),J) . v = FALSE
by XBOOLEAN:def 3;
(Valid ((All x,p) => p),J) . v =
(Valid ('not' ((All x,p) '&' ('not' p))),J) . v
by QC_LANG2:def 2
.=
'not' ((Valid ((All x,p) '&' ('not' p)),J) . v)
by Th20
.=
'not' (((Valid (All x,p),J) . v) '&' ((Valid ('not' p),J) . v))
by Th22
.=
'not' (((Valid (All x,p),J) . v) '&' ('not' ((Valid p,J) . v)))
by Th20
;
then
((Valid (All x,p),J) . v) '&' ('not' ((Valid p,J) . v)) = TRUE
by A1, MARGREL1:41;
then A2:
(
(Valid (All x,p),J) . v = TRUE &
'not' ((Valid p,J) . v) = TRUE )
by MARGREL1:45;
then A3:
(
(Valid (All x,p),J) . v = TRUE &
(Valid p,J) . v = FALSE )
by MARGREL1:41;
(FOR_ALL x,(Valid p,J)) . v = TRUE
by A2, Lm1;
hence
contradiction
by A3, Th37;
:: thesis: verum
end;