let p, q, s be Element of LTLB_WFF ; ('not' (s '&&' q)) => ((p => q) => (p => ('not' s))) is ctaut
let g be Function of LTLB_WFF,BOOLEAN; LTLAXIO1:def 16 (VAL g) . (('not' (s '&&' q)) => ((p => q) => (p => ('not' s)))) = 1
set v = VAL g;
A1:
(VAL g) . TFALSUM = 0
by LTLAXIO1:def 15;
A2:
( (VAL g) . p = 1 or (VAL g) . p = 0 )
by XBOOLEAN:def 3;
A3: (VAL g) . ('not' (s '&&' q)) =
((VAL g) . (s '&&' q)) => ((VAL g) . TFALSUM)
by LTLAXIO1:def 15
.=
(((VAL g) . s) '&' ((VAL g) . q)) => ((VAL g) . TFALSUM)
by LTLAXIO1:31
;
A4:
( (VAL g) . s = 1 or (VAL g) . s = 0 )
by XBOOLEAN:def 3;
A5:
( (VAL g) . q = 1 or (VAL g) . q = 0 )
by XBOOLEAN:def 3;
(VAL g) . ((p => q) => (p => ('not' s))) =
((VAL g) . (p => q)) => ((VAL g) . (p => ('not' s)))
by LTLAXIO1:def 15
.=
(((VAL g) . p) => ((VAL g) . q)) => ((VAL g) . (p => ('not' s)))
by LTLAXIO1:def 15
.=
(((VAL g) . p) => ((VAL g) . q)) => (((VAL g) . p) => ((VAL g) . ('not' s)))
by LTLAXIO1:def 15
.=
(((VAL g) . p) => ((VAL g) . q)) => (((VAL g) . p) => (((VAL g) . s) => ((VAL g) . TFALSUM)))
by LTLAXIO1:def 15
;
hence (VAL g) . (('not' (s '&&' q)) => ((p => q) => (p => ('not' s)))) =
((((VAL g) . s) '&' ((VAL g) . q)) => ((VAL g) . TFALSUM)) => ((((VAL g) . p) => ((VAL g) . q)) => (((VAL g) . p) => (((VAL g) . s) => ((VAL g) . TFALSUM))))
by LTLAXIO1:def 15, A3
.=
1
by A2, A5, A4, A1
;
verum