let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds (a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c))) = I_el Y
let a, b, c be Element of Funcs Y,BOOLEAN ; :: thesis: (a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c))) = I_el Y
consider k3 being Function such that
A1: (a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c))) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: I_el Y = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds ((a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c)))) . x = (I_el Y) . x

proof

let x be Element of Y; :: thesis: ((a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c)))) . x = (I_el Y) . x
A5: ((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 11
.= ('not' ((a 'imp' b) . x)) 'or' (('not' (('not' (b '&' c)) . x)) 'or' (('not' (a '&' c)) . x)) by BVFUNC_1:def 11 ;
A6: ('not' (a '&' c)) . x = (('not' a) 'or' ('not' c)) . x by BVFUNC_1:17
.= (('not' a) . x) 'or' (('not' c) . x) by BVFUNC_1:def 7
.= ('not' (a . x)) 'or' (('not' c) . x) by MARGREL1:def 20
.= ('not' (a . x)) 'or' ('not' (c . x)) by MARGREL1:def 20 ;

now

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:19; :: 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 MARGREL1:41
.= TRUE by BINARITH:19 ;
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:19;
then A7: ((('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 MARGREL1:50;

A8: now

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:19; :: 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 MARGREL1:41
.= TRUE by BINARITH:19 ;
hence ('not' (a . x)) 'or' (a . x) = TRUE ; :: thesis: verum

end;

end;

end;

A9: now

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:19; :: 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 MARGREL1:41
.= TRUE by BINARITH:19 ;
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 20
.= (b . x) '&' (c . x) by MARGREL1:def 21 ;
then A10: ('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 A6, XBOOLEAN:9
.= ((('not' (a . x)) 'or' ('not' (c . x))) 'or' (b . x)) '&' (('not' (a . x)) 'or' (('not' (c . x)) 'or' (c . x))) by BINARITH:20 ;
'not' ((a 'imp' b) . x) = 'not' (('not' (a . x)) 'or' (b . x)) by BVFUNC_1:def 11
.= (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 A10, A7, 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:20
.= (((('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:20
.= ((('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:20 ;
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 A8, A9, BINARITH:19
.= TRUE '&' ((('not' (a . x)) 'or' ('not' (c . x))) 'or' TRUE ) by BINARITH:19
.= TRUE '&' TRUE by BINARITH:19
.= TRUE ;
hence ((a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c)))) . x = (I_el Y) . x by A5, BVFUNC_1:def 14; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence (a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c))) = I_el Y by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum