let I, x be set ; :: thesis: bool (I --> x) = I --> (bool x)

now :: thesis: for i being object st i in I holds
(bool (I --> x)) . i = (I --> (bool x)) . i

let i be object ; :: thesis: ( i in I implies (bool (I --> x)) . i = (I --> (bool x)) . i )
assume A1: i in I ; :: thesis: (bool (I --> x)) . i = (I --> (bool x)) . i
hence (bool (I --> x)) . i = bool ((I --> x) . i) by Def1
.= bool x by A1, FUNCOP_1:7
.= (I --> (bool x)) . i by A1, FUNCOP_1:7 ;
:: thesis: verum

end;

hence bool (I --> x) = I --> (bool x) ; :: thesis: verum