let p, q, r, s be Element of LTLB_WFF ; (q => (p '&&' r)) => ((p => s) => (q => (s '&&' r))) is ctaut
let g be Function of LTLB_WFF,BOOLEAN; LTLAXIO1:def 16 (VAL g) . ((q => (p '&&' r)) => ((p => s) => (q => (s '&&' r)))) = 1
set v = VAL g;
A1:
( (VAL g) . p = 1 or (VAL g) . p = 0 )
by XBOOLEAN:def 3;
A2:
( (VAL g) . r = 1 or (VAL g) . r = 0 )
by XBOOLEAN:def 3;
A3: (VAL g) . ((p => s) => (q => (s '&&' r))) =
((VAL g) . (p => s)) => ((VAL g) . (q => (s '&&' r)))
by LTLAXIO1:def 15
.=
(((VAL g) . p) => ((VAL g) . s)) => ((VAL g) . (q => (s '&&' r)))
by LTLAXIO1:def 15
.=
(((VAL g) . p) => ((VAL g) . s)) => (((VAL g) . q) => ((VAL g) . (s '&&' r)))
by LTLAXIO1:def 15
.=
(((VAL g) . p) => ((VAL g) . s)) => (((VAL g) . q) => (((VAL g) . s) '&' ((VAL g) . r)))
by LTLAXIO1:31
;
A4:
( (VAL g) . q = 1 or (VAL g) . q = 0 )
by XBOOLEAN:def 3;
A5:
( (VAL g) . s = 1 or (VAL g) . s = 0 )
by XBOOLEAN:def 3;
(VAL g) . (q => (p '&&' r)) =
((VAL g) . q) => ((VAL g) . (p '&&' r))
by LTLAXIO1:def 15
.=
((VAL g) . q) => (((VAL g) . p) '&' ((VAL g) . r))
by LTLAXIO1:31
;
hence (VAL g) . ((q => (p '&&' r)) => ((p => s) => (q => (s '&&' r)))) =
(((VAL g) . q) => (((VAL g) . p) '&' ((VAL g) . r))) => ((((VAL g) . p) => ((VAL g) . s)) => (((VAL g) . q) => (((VAL g) . s) '&' ((VAL g) . r))))
by LTLAXIO1:def 15, A3
.=
1
by A1, A2, A5, A4
;
verum