let p, q, r be Element of LTLB_WFF ; :: thesis: (p => q) => ((p => r) => ((r => p) => (r => q))) is ctaut
let g be Function of LTLB_WFF,BOOLEAN; :: according to LTLAXIO1:def 16 :: thesis: (VAL g) . ((p => q) => ((p => r) => ((r => p) => (r => q)))) = 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 => r) => ((r => p) => (r => q))) = ((VAL g) . (p => r)) => ((VAL g) . ((r => p) => (r => q))) by LTLAXIO1:def 15
.= (((VAL g) . p) => ((VAL g) . r)) => ((VAL g) . ((r => p) => (r => q))) by LTLAXIO1:def 15
.= (((VAL g) . p) => ((VAL g) . r)) => (((VAL g) . (r => p)) => ((VAL g) . (r => q))) by LTLAXIO1:def 15
.= (((VAL g) . p) => ((VAL g) . r)) => ((((VAL g) . r) => ((VAL g) . p)) => ((VAL g) . (r => q))) by LTLAXIO1:def 15
.= (((VAL g) . p) => ((VAL g) . r)) => ((((VAL g) . r) => ((VAL g) . p)) => (((VAL g) . r) => ((VAL g) . q))) by LTLAXIO1:def 15 ;
A4: ( (VAL g) . q = 1 or (VAL g) . q = 0 ) by XBOOLEAN:def 3;
(VAL g) . (p => q) = ((VAL g) . p) => ((VAL g) . q) by LTLAXIO1:def 15;
hence (VAL g) . ((p => q) => ((p => r) => ((r => p) => (r => q)))) = (((VAL g) . p) => ((VAL g) . q)) => ((((VAL g) . p) => ((VAL g) . r)) => ((((VAL g) . r) => ((VAL g) . p)) => (((VAL g) . r) => ((VAL g) . q)))) by LTLAXIO1:def 15, A3
.= 1 by A1, A2, A4 ;
:: thesis: verum