let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c)) = I_el Y
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c)) = I_el Y
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c))) . x = (I_el Y) . x
A1: ((a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c))) . x = ('not' ((a 'imp' b) . x)) 'or' (((b 'imp' c) 'imp' (a 'imp' c)) . x) by BVFUNC_1:def 8
.= ('not' ((a 'imp' b) . x)) 'or' (('not' ((b 'imp' c) . x)) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 8 ;
A2: 'not' ((b 'imp' c) . x) = 'not' (('not' (b . x)) 'or' (c . x)) by BVFUNC_1:def 8
.= (b . x) '&' ('not' (c . x)) ;

A3: now :: thesis: ( ( b . x = TRUE & ('not' (b . x)) 'or' (b . x) = TRUE ) or ( b . x = FALSE & ('not' (b . x)) 'or' (b . x) = TRUE ) )

per cases ( b . x = TRUE or b . x = FALSE ) by XBOOLEAN:def 3;

case b . x = TRUE ; :: thesis: ('not' (b . x)) 'or' (b . x) = TRUE

hence ('not' (b . x)) 'or' (b . x) = TRUE by BINARITH:10; :: thesis: verum

end;

case b . x = FALSE ; :: thesis: ('not' (b . x)) 'or' (b . x) = TRUE

then ('not' (b . x)) 'or' (b . x) = TRUE 'or' FALSE by MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ('not' (b . x)) 'or' (b . x) = TRUE ; :: thesis: verum

end;

end;

end;

A4: (a 'imp' c) . x = ('not' (a . x)) 'or' (c . x) by BVFUNC_1:def 8;

A5: now :: thesis: ( ( c . x = TRUE & ('not' (c . x)) 'or' (c . x) = TRUE ) or ( c . x = FALSE & ('not' (c . x)) 'or' (c . x) = TRUE ) )

per cases ( c . x = TRUE or c . x = FALSE ) by XBOOLEAN:def 3;

case c . x = TRUE ; :: thesis: ('not' (c . x)) 'or' (c . x) = TRUE

hence ('not' (c . x)) 'or' (c . x) = TRUE by BINARITH:10; :: thesis: verum

end;

case c . x = FALSE ; :: thesis: ('not' (c . x)) 'or' (c . x) = TRUE

then ('not' (c . x)) 'or' (c . x) = TRUE 'or' FALSE by MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ('not' (c . x)) 'or' (c . x) = TRUE ; :: thesis: verum

end;

end;

end;

A6: now :: thesis: ( ( a . x = TRUE & ('not' (a . x)) 'or' (a . x) = TRUE ) or ( a . x = FALSE & ('not' (a . x)) 'or' (a . x) = TRUE ) )

per cases ( a . x = TRUE or a . x = FALSE ) by XBOOLEAN:def 3;

case a . x = TRUE ; :: thesis: ('not' (a . x)) 'or' (a . x) = TRUE

hence ('not' (a . x)) 'or' (a . x) = TRUE by BINARITH:10; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: ('not' (a . x)) 'or' (a . x) = TRUE

then ('not' (a . x)) 'or' (a . x) = TRUE 'or' FALSE by MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ('not' (a . x)) 'or' (a . x) = TRUE ; :: thesis: verum

end;

end;

end;

'not' ((a 'imp' b) . x) = 'not' (('not' (a . x)) 'or' (b . x)) by BVFUNC_1:def 8
.= (a . x) '&' ('not' (b . x)) ;
then ('not' ((a 'imp' b) . x)) 'or' (('not' ((b 'imp' c) . x)) 'or' ((a 'imp' c) . x)) = ((((b . x) '&' ('not' (c . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (a . x)) '&' ((((b . x) '&' ('not' (c . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' ('not' (b . x))) by A2, A4, XBOOLEAN:9
.= (((b . x) '&' ('not' (c . x))) 'or' (((c . x) 'or' ('not' (a . x))) 'or' (a . x))) '&' ((((b . x) '&' ('not' (c . x))) 'or' (('not' (a . x)) 'or' (c . x))) 'or' ('not' (b . x))) by BINARITH:11
.= (((b . x) '&' ('not' (c . x))) 'or' ((c . x) 'or' (('not' (a . x)) 'or' (a . x)))) '&' (((('not' (a . x)) 'or' (c . x)) 'or' ((b . x) '&' ('not' (c . x)))) 'or' ('not' (b . x))) by BINARITH:11
.= (((b . x) '&' ('not' (c . x))) 'or' ((c . x) 'or' (('not' (a . x)) 'or' (a . x)))) '&' ((('not' (a . x)) 'or' (c . x)) 'or' ((('not' (c . x)) '&' (b . x)) 'or' ('not' (b . x)))) by BINARITH:11
.= (((b . x) '&' ('not' (c . x))) 'or' ((c . x) 'or' (('not' (a . x)) 'or' (a . x)))) '&' ((('not' (a . x)) 'or' (c . x)) 'or' ((('not' (b . x)) 'or' ('not' (c . x))) '&' (('not' (b . x)) 'or' (b . x)))) by XBOOLEAN:9
.= (((b . x) '&' ('not' (c . x))) 'or' TRUE) '&' ((('not' (a . x)) 'or' (c . x)) 'or' ((('not' (b . x)) 'or' ('not' (c . x))) '&' TRUE)) by A3, A6, BINARITH:10
.= TRUE '&' ((('not' (a . x)) 'or' (c . x)) 'or' ((('not' (b . x)) 'or' ('not' (c . x))) '&' TRUE)) by BINARITH:10
.= (('not' (a . x)) 'or' (c . x)) 'or' (TRUE '&' (('not' (b . x)) 'or' ('not' (c . x)))) by MARGREL1:14
.= (('not' (a . x)) 'or' (c . x)) 'or' (('not' (c . x)) 'or' ('not' (b . x))) by MARGREL1:14
.= ((('not' (a . x)) 'or' (c . x)) 'or' ('not' (c . x))) 'or' ('not' (b . x)) by BINARITH:11
.= (('not' (a . x)) 'or' TRUE) 'or' ('not' (b . x)) by A5, BINARITH:11
.= TRUE 'or' ('not' (b . x)) by BINARITH:10
.= TRUE by BINARITH:10 ;
hence ((a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c))) . x = (I_el Y) . x by A1, BVFUNC_1:def 11; :: thesis: verum