let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c))) = I_el Y
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c))) = I_el Y
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c)))) . x = (I_el Y) . x
A1: ((a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c)))) . x = ('not' ((a 'imp' b) . x)) 'or' ((('not' (b '&' c)) 'imp' ('not' (a '&' c))) . x) by BVFUNC_1:def 8
.= ('not' ((a 'imp' b) . x)) 'or' (('not' (('not' (b '&' c)) . x)) 'or' (('not' (a '&' c)) . x)) by BVFUNC_1:def 8 ;
A2: ('not' (a '&' c)) . x = (('not' a) 'or' ('not' c)) . x by BVFUNC_1:14
.= (('not' a) . x) 'or' (('not' c) . x) by BVFUNC_1:def 4
.= ('not' (a . x)) 'or' (('not' c) . x) by MARGREL1:def 19
.= ('not' (a . x)) 'or' ('not' (c . x)) by MARGREL1:def 19 ;

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;

then ('not' (a . x)) 'or' (('not' (c . x)) 'or' (c . x)) = TRUE by BINARITH:10;
then A3: ((('not' (a . x)) 'or' ('not' (c . x))) 'or' (b . x)) '&' (('not' (a . x)) 'or' (('not' (c . x)) 'or' (c . x))) = (('not' (a . x)) 'or' ('not' (c . x))) 'or' (b . x) by MARGREL1:14;

A4: 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;

A5: 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;

'not' (('not' (b '&' c)) . x) = 'not' ('not' ((b '&' c) . x)) by MARGREL1:def 19
.= (b . x) '&' (c . x) by MARGREL1:def 20 ;
then A6: ('not' (('not' (b '&' c)) . x)) 'or' (('not' (a '&' c)) . x) = ((('not' (a . x)) 'or' ('not' (c . x))) 'or' (b . x)) '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' (c . x)) by A2, XBOOLEAN:9
.= ((('not' (a . x)) 'or' ('not' (c . x))) 'or' (b . x)) '&' (('not' (a . x)) 'or' (('not' (c . x)) 'or' (c . x))) by BINARITH:11 ;
'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' (('not' (b '&' c)) . x)) 'or' (('not' (a '&' c)) . x)) = (((('not' (a . x)) 'or' ('not' (c . x))) 'or' (b . x)) 'or' (a . x)) '&' (((('not' (a . x)) 'or' ('not' (c . x))) 'or' (b . x)) 'or' ('not' (b . x))) by A6, A3, XBOOLEAN:9
.= (((('not' (a . x)) 'or' ('not' (c . x))) 'or' (b . x)) 'or' (a . x)) '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' ((b . x) 'or' ('not' (b . x)))) by BINARITH:11
.= (((('not' (c . x)) 'or' ('not' (a . x))) 'or' (a . x)) 'or' (b . x)) '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' ((b . x) 'or' ('not' (b . x)))) by BINARITH:11
.= ((('not' (c . x)) 'or' (('not' (a . x)) 'or' (a . x))) 'or' (b . x)) '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' ((b . x) 'or' ('not' (b . x)))) by BINARITH:11 ;
then ('not' ((a 'imp' b) . x)) 'or' (('not' (('not' (b '&' c)) . x)) 'or' (('not' (a '&' c)) . x)) = (TRUE 'or' (b . x)) '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' TRUE) by A4, A5, BINARITH:10
.= TRUE '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' TRUE) by BINARITH:10
.= TRUE '&' TRUE by BINARITH:10
.= TRUE ;
hence ((a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c)))) . x = (I_el Y) . x by A1, BVFUNC_1:def 11; :: thesis: verum