let X be BCI-algebra; :: thesis: ( X is p-Semisimple iff for x, y being Element of X holds x \ (y ` ) = y \ (x ` ) )
thus ( X is p-Semisimple implies for x, y being Element of X holds x \ (y ` ) = y \ (x ` ) ) by Th57; :: thesis: ( ( for x, y being Element of X holds x \ (y ` ) = y \ (x ` ) ) implies X is p-Semisimple )
assume A1: for x, y being Element of X holds x \ (y ` ) = y \ (x ` ) ; :: thesis: X is p-Semisimple

now

let x be Element of X; :: thesis: x = (x ` ) `
x \ ((0. X) ` ) = (x ` ) ` by A1;
then x \ (0. X) = (x ` ) ` by Th2;
hence x = (x ` ) ` by Th2; :: thesis: verum

end;

hence X is p-Semisimple by Th54; :: thesis: verum