:: BVFUNC_8 semantic presentation begin theorem :: BVFUNC_8:1 for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds a 'imp' ((b '&' c) '&' d) = ((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d) proof let Y be non empty set ; ::_thesis: for a, b, c, d being Function of Y,BOOLEAN holds a 'imp' ((b '&' c) '&' d) = ((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d) let a, b, c, d be Function of Y,BOOLEAN; ::_thesis: a 'imp' ((b '&' c) '&' d) = ((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'imp' ((b '&' c) '&' d)) . x = (((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)) . x (((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)) . x = (((a 'imp' b) '&' (a 'imp' c)) . x) '&' ((a 'imp' d) . x) by MARGREL1:def_20 .= (((a 'imp' b) . x) '&' ((a 'imp' c) . x)) '&' ((a 'imp' d) . x) by MARGREL1:def_20 .= ((('not' (a . x)) 'or' (b . x)) '&' ((a 'imp' c) . x)) '&' ((a 'imp' d) . x) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' (b . x)) '&' (('not' (a . x)) 'or' (c . x))) '&' ((a 'imp' d) . x) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' (b . x)) '&' (('not' (a . x)) 'or' (c . x))) '&' (('not' (a . x)) 'or' (d . x)) by BVFUNC_1:def_8 .= (('not' (a . x)) 'or' ((b . x) '&' (c . x))) '&' (('not' (a . x)) 'or' (d . x)) by XBOOLEAN:9 .= ('not' (a . x)) 'or' (((b . x) '&' (c . x)) '&' (d . x)) by XBOOLEAN:9 .= ('not' (a . x)) 'or' (((b '&' c) . x) '&' (d . x)) by MARGREL1:def_20 .= ('not' (a . x)) 'or' (((b '&' c) '&' d) . x) by MARGREL1:def_20 .= (a 'imp' ((b '&' c) '&' d)) . x by BVFUNC_1:def_8 ; hence (a 'imp' ((b '&' c) '&' d)) . x = (((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:2 for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds a 'imp' ((b 'or' c) 'or' d) = ((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d) proof let Y be non empty set ; ::_thesis: for a, b, c, d being Function of Y,BOOLEAN holds a 'imp' ((b 'or' c) 'or' d) = ((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d) let a, b, c, d be Function of Y,BOOLEAN; ::_thesis: a 'imp' ((b 'or' c) 'or' d) = ((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'imp' ((b 'or' c) 'or' d)) . x = (((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d)) . x (((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d)) . x = (((a 'imp' b) 'or' (a 'imp' c)) . x) 'or' ((a 'imp' d) . x) by BVFUNC_1:def_4 .= (((a 'imp' b) . x) 'or' ((a 'imp' c) . x)) 'or' ((a 'imp' d) . x) by BVFUNC_1:def_4 .= ((('not' (a . x)) 'or' (b . x)) 'or' ((a 'imp' c) . x)) 'or' ((a 'imp' d) . x) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' (b . x)) 'or' (('not' (a . x)) 'or' (c . x))) 'or' ((a 'imp' d) . x) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' (b . x)) 'or' (('not' (a . x)) 'or' (c . x))) 'or' (('not' (a . x)) 'or' (d . x)) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' (('not' (a . x)) 'or' (b . x))) 'or' (c . x)) 'or' (('not' (a . x)) 'or' (d . x)) by BINARITH:11 .= (((('not' (a . x)) 'or' ('not' (a . x))) 'or' (b . x)) 'or' (c . x)) 'or' (('not' (a . x)) 'or' (d . x)) by BINARITH:11 .= (('not' (a . x)) 'or' ((b . x) 'or' (c . x))) 'or' (('not' (a . x)) 'or' (d . x)) by BINARITH:11 .= (('not' (a . x)) 'or' ((b 'or' c) . x)) 'or' (('not' (a . x)) 'or' (d . x)) by BVFUNC_1:def_4 .= (('not' (a . x)) 'or' (('not' (a . x)) 'or' ((b 'or' c) . x))) 'or' (d . x) by BINARITH:11 .= ((('not' (a . x)) 'or' ('not' (a . x))) 'or' ((b 'or' c) . x)) 'or' (d . x) by BINARITH:11 .= ('not' (a . x)) 'or' (((b 'or' c) . x) 'or' (d . x)) by BINARITH:11 .= ('not' (a . x)) 'or' (((b 'or' c) 'or' d) . x) by BVFUNC_1:def_4 .= (a 'imp' ((b 'or' c) 'or' d)) . x by BVFUNC_1:def_8 ; hence (a 'imp' ((b 'or' c) 'or' d)) . x = (((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:3 for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d) proof let Y be non empty set ; ::_thesis: for a, b, c, d being Function of Y,BOOLEAN holds ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d) let a, b, c, d be Function of Y,BOOLEAN; ::_thesis: ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (((a '&' b) '&' c) 'imp' d) . x = (((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x (((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x = (((a 'imp' d) 'or' (b 'imp' d)) . x) 'or' ((c 'imp' d) . x) by BVFUNC_1:def_4 .= (((a 'imp' d) . x) 'or' ((b 'imp' d) . x)) 'or' ((c 'imp' d) . x) by BVFUNC_1:def_4 .= ((('not' (a . x)) 'or' (d . x)) 'or' ((b 'imp' d) . x)) 'or' ((c 'imp' d) . x) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' (d . x)) 'or' (('not' (b . x)) 'or' (d . x))) 'or' ((c 'imp' d) . x) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' (d . x)) 'or' (('not' (b . x)) 'or' (d . x))) 'or' (('not' (c . x)) 'or' (d . x)) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' ((d . x) 'or' ('not' (b . x)))) 'or' (d . x)) 'or' (('not' (c . x)) 'or' (d . x)) by BINARITH:11 .= (((('not' (a . x)) 'or' ('not' (b . x))) 'or' (d . x)) 'or' (d . x)) 'or' (('not' (c . x)) 'or' (d . x)) by BINARITH:11 .= ((('not' (a . x)) 'or' ('not' (b . x))) 'or' ((d . x) 'or' (d . x))) 'or' (('not' (c . x)) 'or' (d . x)) by BINARITH:11 .= (('not' ((a . x) '&' (b . x))) 'or' ((d . x) 'or' ('not' (c . x)))) 'or' (d . x) by BINARITH:11 .= ((('not' ((a . x) '&' (b . x))) 'or' ('not' (c . x))) 'or' (d . x)) 'or' (d . x) by BINARITH:11 .= (('not' ((a . x) '&' (b . x))) 'or' ('not' (c . x))) 'or' ((d . x) 'or' (d . x)) by BINARITH:11 .= ('not' (((a '&' b) . x) '&' (c . x))) 'or' (d . x) by MARGREL1:def_20 .= ('not' (((a '&' b) '&' c) . x)) 'or' (d . x) by MARGREL1:def_20 .= (((a '&' b) '&' c) 'imp' d) . x by BVFUNC_1:def_8 ; hence (((a '&' b) '&' c) 'imp' d) . x = (((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:4 for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds ((a 'or' b) 'or' c) 'imp' d = ((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d) proof let Y be non empty set ; ::_thesis: for a, b, c, d being Function of Y,BOOLEAN holds ((a 'or' b) 'or' c) 'imp' d = ((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d) let a, b, c, d be Function of Y,BOOLEAN; ::_thesis: ((a 'or' b) 'or' c) 'imp' d = ((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (((a 'or' b) 'or' c) 'imp' d) . x = (((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)) . x (((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)) . x = (((a 'imp' d) '&' (b 'imp' d)) . x) '&' ((c 'imp' d) . x) by MARGREL1:def_20 .= (((a 'imp' d) . x) '&' ((b 'imp' d) . x)) '&' ((c 'imp' d) . x) by MARGREL1:def_20 .= ((('not' (a . x)) 'or' (d . x)) '&' ((b 'imp' d) . x)) '&' ((c 'imp' d) . x) by BVFUNC_1:def_8 .= ((('not' (a . x)) 'or' (d . x)) '&' (('not' (b . x)) 'or' (d . x))) '&' ((c 'imp' d) . x) by BVFUNC_1:def_8 .= (((d . x) 'or' ('not' (a . x))) '&' (('not' (b . x)) 'or' (d . x))) '&' (('not' (c . x)) 'or' (d . x)) by BVFUNC_1:def_8 .= (('not' ((a . x) 'or' (b . x))) 'or' (d . x)) '&' (('not' (c . x)) 'or' (d . x)) by XBOOLEAN:9 .= ((d . x) 'or' ('not' ((a 'or' b) . x))) '&' (('not' (c . x)) 'or' (d . x)) by BVFUNC_1:def_4 .= ('not' (((a 'or' b) . x) 'or' (c . x))) 'or' (d . x) by XBOOLEAN:9 .= ('not' (((a 'or' b) 'or' c) . x)) 'or' (d . x) by BVFUNC_1:def_4 .= (((a 'or' b) 'or' c) 'imp' d) . x by BVFUNC_1:def_8 ; hence (((a 'or' b) 'or' c) 'imp' d) . x = (((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:5 for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a) = ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c) proof let Y be non empty set ; ::_thesis: for a, b, c being Function of Y,BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a) = ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c) let a, b, c be Function of Y,BOOLEAN; ::_thesis: ((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a) = ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) . x = (((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c)) . x ((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a) = ((('not' a) 'or' b) '&' (b 'imp' c)) '&' (c 'imp' a) by BVFUNC_4:8 .= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (c 'imp' a) by BVFUNC_4:8 .= ((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (('not' c) 'or' a) by BVFUNC_4:8 .= ((('not' a) '&' (('not' b) 'or' c)) 'or' (b '&' (('not' b) 'or' c))) '&' (('not' c) 'or' a) by BVFUNC_1:12 .= ((('not' a) '&' (('not' b) 'or' c)) 'or' ((b '&' ('not' b)) 'or' (b '&' c))) '&' (('not' c) 'or' a) by BVFUNC_1:12 .= ((('not' a) '&' (('not' b) 'or' c)) 'or' ((O_el Y) 'or' (b '&' c))) '&' (('not' c) 'or' a) by BVFUNC_4:5 .= ((('not' a) '&' (('not' b) 'or' c)) 'or' (b '&' c)) '&' (('not' c) 'or' a) by BVFUNC_1:9 .= ((('not' a) 'or' (b '&' c)) '&' ((('not' b) 'or' c) 'or' (b '&' c))) '&' (('not' c) 'or' a) by BVFUNC_1:11 .= (((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((('not' b) 'or' c) 'or' (b '&' c))) '&' (('not' c) 'or' a) by BVFUNC_1:11 .= (((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' (((('not' b) 'or' c) 'or' b) '&' ((('not' b) 'or' c) 'or' c))) '&' (('not' c) 'or' a) by BVFUNC_1:11 .= (((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((c 'or' (('not' b) 'or' b)) '&' ((('not' b) 'or' c) 'or' c))) '&' (('not' c) 'or' a) by BVFUNC_1:8 .= (((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((c 'or' (I_el Y)) '&' ((('not' b) 'or' c) 'or' c))) '&' (('not' c) 'or' a) by BVFUNC_4:6 .= (((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((I_el Y) '&' ((('not' b) 'or' c) 'or' c))) '&' (('not' c) 'or' a) by BVFUNC_1:10 .= (((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((('not' b) 'or' c) 'or' c)) '&' (('not' c) 'or' a) by BVFUNC_1:6 .= (((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' (('not' b) 'or' (c 'or' c))) '&' (('not' c) 'or' a) by BVFUNC_1:8 .= ((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((('not' b) 'or' c) '&' (('not' c) 'or' a)) by BVFUNC_1:4 .= ((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((('not' b) '&' (('not' c) 'or' a)) 'or' (c '&' (('not' c) 'or' a))) by BVFUNC_1:12 .= ((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((('not' b) '&' (('not' c) 'or' a)) 'or' ((c '&' ('not' c)) 'or' (c '&' a))) by BVFUNC_1:12 .= ((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((('not' b) '&' (('not' c) 'or' a)) 'or' ((O_el Y) 'or' (c '&' a))) by BVFUNC_4:5 .= ((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((('not' b) '&' (('not' c) 'or' a)) 'or' (c '&' a)) by BVFUNC_1:9 .= ((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' ((('not' b) 'or' (c '&' a)) '&' ((('not' c) 'or' a) 'or' (c '&' a))) by BVFUNC_1:11 .= ((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' (((('not' b) 'or' c) '&' (('not' b) 'or' a)) '&' ((('not' c) 'or' a) 'or' (c '&' a))) by BVFUNC_1:11 .= ((('not' a) 'or' c) '&' (('not' a) 'or' b)) '&' (((('not' b) 'or' a) '&' (('not' b) 'or' c)) '&' (((('not' c) 'or' a) 'or' c) '&' ((('not' c) 'or' a) 'or' a))) by BVFUNC_1:11 .= ((('not' a) 'or' c) '&' (('not' a) 'or' b)) '&' (((('not' b) 'or' a) '&' (('not' b) 'or' c)) '&' ((a 'or' (('not' c) 'or' c)) '&' ((('not' c) 'or' a) 'or' a))) by BVFUNC_1:8 .= ((('not' a) 'or' c) '&' (('not' a) 'or' b)) '&' (((('not' b) 'or' a) '&' (('not' b) 'or' c)) '&' ((a 'or' (I_el Y)) '&' ((('not' c) 'or' a) 'or' a))) by BVFUNC_4:6 .= ((('not' a) 'or' c) '&' (('not' a) 'or' b)) '&' (((('not' b) 'or' a) '&' (('not' b) 'or' c)) '&' ((I_el Y) '&' ((('not' c) 'or' a) 'or' a))) by BVFUNC_1:10 .= ((('not' a) 'or' c) '&' (('not' a) 'or' b)) '&' (((('not' b) 'or' a) '&' (('not' b) 'or' c)) '&' ((('not' c) 'or' a) 'or' a)) by BVFUNC_1:6 .= ((('not' a) 'or' c) '&' (('not' a) 'or' b)) '&' (((('not' b) 'or' a) '&' (('not' b) 'or' c)) '&' (('not' c) 'or' (a 'or' a))) by BVFUNC_1:8 .= (((('not' a) 'or' c) '&' (('not' a) 'or' b)) '&' ((('not' b) 'or' a) '&' (('not' b) 'or' c))) '&' (('not' c) 'or' a) by BVFUNC_1:4 .= ((((('not' a) 'or' b) '&' (('not' a) 'or' c)) '&' (('not' b) 'or' a)) '&' (('not' b) 'or' c)) '&' (('not' c) 'or' a) by BVFUNC_1:4 .= ((((('not' b) 'or' a) '&' (('not' a) 'or' c)) '&' (('not' a) 'or' b)) '&' (('not' b) 'or' c)) '&' (('not' c) 'or' a) by BVFUNC_1:4 .= (((('not' b) 'or' a) '&' (('not' a) 'or' c)) '&' ((('not' a) 'or' b) '&' (('not' b) 'or' c))) '&' (('not' c) 'or' a) by BVFUNC_1:4 .= ((('not' b) 'or' a) '&' (('not' a) 'or' c)) '&' (((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (('not' c) 'or' a)) by BVFUNC_1:4 .= ((('not' b) 'or' a) '&' (((('not' a) 'or' b) '&' (('not' b) 'or' c)) '&' (('not' c) 'or' a))) '&' (('not' a) 'or' c) by BVFUNC_1:4 .= ((((a 'imp' b) '&' (('not' b) 'or' c)) '&' (('not' c) 'or' a)) '&' (('not' b) 'or' a)) '&' (('not' a) 'or' c) by BVFUNC_4:8 .= ((((a 'imp' b) '&' (b 'imp' c)) '&' (('not' c) 'or' a)) '&' (('not' b) 'or' a)) '&' (('not' a) 'or' c) by BVFUNC_4:8 .= ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (('not' b) 'or' a)) '&' (('not' a) 'or' c) by BVFUNC_4:8 .= ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (('not' a) 'or' c) by BVFUNC_4:8 .= ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c) by BVFUNC_4:8 ; hence (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) . x = (((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:6 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a = (a '&' b) 'or' (a '&' ('not' b)) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a = (a '&' b) 'or' (a '&' ('not' b)) let a, b be Function of Y,BOOLEAN; ::_thesis: a = (a '&' b) 'or' (a '&' ('not' b)) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: a . x = ((a '&' b) 'or' (a '&' ('not' b))) . x ((a '&' b) 'or' (a '&' ('not' b))) . x = (a '&' (b 'or' ('not' b))) . x by BVFUNC_1:12 .= (a '&' (I_el Y)) . x by BVFUNC_4:6 .= a . x by BVFUNC_1:6 ; hence a . x = ((a '&' b) 'or' (a '&' ('not' b))) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:7 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a = (a 'or' b) '&' (a 'or' ('not' b)) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a = (a 'or' b) '&' (a 'or' ('not' b)) let a, b be Function of Y,BOOLEAN; ::_thesis: a = (a 'or' b) '&' (a 'or' ('not' b)) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: a . x = ((a 'or' b) '&' (a 'or' ('not' b))) . x ((a 'or' b) '&' (a 'or' ('not' b))) . x = (a 'or' (b '&' ('not' b))) . x by BVFUNC_1:11 .= (a 'or' (O_el Y)) . x by BVFUNC_4:5 .= a . x by BVFUNC_1:9 ; hence a . x = ((a 'or' b) '&' (a 'or' ('not' b))) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:8 for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a = ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c)) proof let Y be non empty set ; ::_thesis: for a, b, c being Function of Y,BOOLEAN holds a = ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c)) let a, b, c be Function of Y,BOOLEAN; ::_thesis: a = ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c)) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: a . x = (((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x (((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x = ((((a '&' b) '&' (c 'or' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x by BVFUNC_1:12 .= ((((a '&' b) '&' (I_el Y)) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x by BVFUNC_4:6 .= (((a '&' b) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x by BVFUNC_1:6 .= ((a '&' b) 'or' (((a '&' ('not' b)) '&' c) 'or' ((a '&' ('not' b)) '&' ('not' c)))) . x by BVFUNC_1:8 .= ((a '&' b) 'or' ((a '&' ('not' b)) '&' (c 'or' ('not' c)))) . x by BVFUNC_1:12 .= ((a '&' b) 'or' ((a '&' ('not' b)) '&' (I_el Y))) . x by BVFUNC_4:6 .= ((a '&' b) 'or' (a '&' ('not' b))) . x by BVFUNC_1:6 .= (a '&' (b 'or' ('not' b))) . x by BVFUNC_1:12 .= (a '&' (I_el Y)) . x by BVFUNC_4:6 .= a . x by BVFUNC_1:6 ; hence a . x = (((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:9 for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a = ((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ('not' c))) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c)) proof let Y be non empty set ; ::_thesis: for a, b, c being Function of Y,BOOLEAN holds a = ((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ('not' c))) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c)) let a, b, c be Function of Y,BOOLEAN; ::_thesis: a = ((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ('not' c))) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c)) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: a . x = (((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ('not' c))) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c))) . x (((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ('not' c))) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c))) . x = ((((a 'or' b) 'or' (c '&' ('not' c))) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c))) . x by BVFUNC_1:11 .= ((((a 'or' b) 'or' (O_el Y)) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c))) . x by BVFUNC_4:5 .= (((a 'or' b) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c))) . x by BVFUNC_1:9 .= ((a 'or' b) '&' (((a 'or' ('not' b)) 'or' c) '&' ((a 'or' ('not' b)) 'or' ('not' c)))) . x by BVFUNC_1:4 .= ((a 'or' b) '&' ((a 'or' ('not' b)) 'or' (c '&' ('not' c)))) . x by BVFUNC_1:11 .= ((a 'or' b) '&' ((a 'or' ('not' b)) 'or' (O_el Y))) . x by BVFUNC_4:5 .= ((a 'or' b) '&' (a 'or' ('not' b))) . x by BVFUNC_1:9 .= (a 'or' (b '&' ('not' b))) . x by BVFUNC_1:11 .= (a 'or' (O_el Y)) . x by BVFUNC_4:5 .= a . x by BVFUNC_1:9 ; hence a . x = (((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ('not' c))) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c))) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:10 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' b = a '&' (('not' a) 'or' b) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a '&' b = a '&' (('not' a) 'or' b) let a, b be Function of Y,BOOLEAN; ::_thesis: a '&' b = a '&' (('not' a) 'or' b) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a '&' b) . x = (a '&' (('not' a) 'or' b)) . x (a '&' (('not' a) 'or' b)) . x = (a . x) '&' ((('not' a) 'or' b) . x) by MARGREL1:def_20 .= (a . x) '&' ((('not' a) . x) 'or' (b . x)) by BVFUNC_1:def_4 .= ((a . x) '&' (('not' a) . x)) 'or' ((a . x) '&' (b . x)) by XBOOLEAN:8 .= ((a . x) '&' ('not' (a . x))) 'or' ((a . x) '&' (b . x)) by MARGREL1:def_19 .= FALSE 'or' ((a . x) '&' (b . x)) by XBOOLEAN:138 .= (a . x) '&' (b . x) by BINARITH:3 .= (a '&' b) . x by MARGREL1:def_20 ; hence (a '&' b) . x = (a '&' (('not' a) 'or' b)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:11 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'or' b = a 'or' (('not' a) '&' b) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a 'or' b = a 'or' (('not' a) '&' b) let a, b be Function of Y,BOOLEAN; ::_thesis: a 'or' b = a 'or' (('not' a) '&' b) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'or' b) . x = (a 'or' (('not' a) '&' b)) . x (a 'or' (('not' a) '&' b)) . x = (a . x) 'or' ((('not' a) '&' b) . x) by BVFUNC_1:def_4 .= (a . x) 'or' ((('not' a) . x) '&' (b . x)) by MARGREL1:def_20 .= ((a . x) 'or' (('not' a) . x)) '&' ((a . x) 'or' (b . x)) by XBOOLEAN:9 .= ((a . x) 'or' ('not' (a . x))) '&' ((a . x) 'or' (b . x)) by MARGREL1:def_19 .= TRUE '&' ((a . x) 'or' (b . x)) by XBOOLEAN:102 .= (a . x) 'or' (b . x) by MARGREL1:14 .= (a 'or' b) . x by BVFUNC_1:def_4 ; hence (a 'or' b) . x = (a 'or' (('not' a) '&' b)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:12 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' b = 'not' (a 'eqv' b) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a 'xor' b = 'not' (a 'eqv' b) let a, b be Function of Y,BOOLEAN; ::_thesis: a 'xor' b = 'not' (a 'eqv' b) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'xor' b) . x = ('not' (a 'eqv' b)) . x (a 'xor' b) . x = ('not' ('not' ((('not' a) '&' b) 'or' (a '&' ('not' b))))) . x by BVFUNC_4:9 .= ('not' (('not' (('not' a) '&' b)) '&' ('not' (a '&' ('not' b))))) . x by BVFUNC_1:13 .= ('not' ((('not' ('not' a)) 'or' ('not' b)) '&' ('not' (a '&' ('not' b))))) . x by BVFUNC_1:14 .= ('not' ((a 'or' ('not' b)) '&' (('not' a) 'or' ('not' ('not' b))))) . x by BVFUNC_1:14 .= ('not' ((b 'imp' a) '&' (('not' a) 'or' b))) . x by BVFUNC_4:8 .= ('not' ((b 'imp' a) '&' (a 'imp' b))) . x by BVFUNC_4:8 .= ('not' (a 'eqv' b)) . x by BVFUNC_4:7 ; hence (a 'xor' b) . x = ('not' (a 'eqv' b)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:13 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' b = (a 'or' b) '&' (('not' a) 'or' ('not' b)) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a 'xor' b = (a 'or' b) '&' (('not' a) 'or' ('not' b)) let a, b be Function of Y,BOOLEAN; ::_thesis: a 'xor' b = (a 'or' b) '&' (('not' a) 'or' ('not' b)) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'xor' b) . x = ((a 'or' b) '&' (('not' a) 'or' ('not' b))) . x ((a 'or' b) '&' (('not' a) 'or' ('not' b))) . x = ((a 'or' b) . x) '&' ((('not' a) 'or' ('not' b)) . x) by MARGREL1:def_20 .= ((a . x) 'or' (b . x)) '&' ((('not' a) 'or' ('not' b)) . x) by BVFUNC_1:def_4 .= ((a . x) 'or' (b . x)) '&' ((('not' a) . x) 'or' (('not' b) . x)) by BVFUNC_1:def_4 .= ((('not' a) . x) '&' ((a . x) 'or' (b . x))) 'or' (((a . x) 'or' (b . x)) '&' (('not' b) . x)) by XBOOLEAN:8 .= (((('not' a) . x) '&' (a . x)) 'or' ((('not' a) . x) '&' (b . x))) 'or' ((('not' b) . x) '&' ((a . x) 'or' (b . x))) by XBOOLEAN:8 .= (((('not' a) . x) '&' (a . x)) 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' ((('not' b) . x) '&' (b . x))) by XBOOLEAN:8 .= ((('not' (a . x)) '&' (a . x)) 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' ((('not' b) . x) '&' (b . x))) by MARGREL1:def_19 .= ((('not' (a . x)) '&' (a . x)) 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' (('not' (b . x)) '&' (b . x))) by MARGREL1:def_19 .= (FALSE 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' (('not' (b . x)) '&' (b . x))) by XBOOLEAN:138 .= (FALSE 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' FALSE) by XBOOLEAN:138 .= ((('not' a) . x) '&' (b . x)) 'or' (((('not' b) . x) '&' (a . x)) 'or' FALSE) by BINARITH:3 .= ((('not' a) . x) '&' (b . x)) 'or' ((a . x) '&' (('not' b) . x)) by BINARITH:3 .= ((('not' a) '&' b) . x) 'or' ((a . x) '&' (('not' b) . x)) by MARGREL1:def_20 .= ((('not' a) '&' b) . x) 'or' ((a '&' ('not' b)) . x) by MARGREL1:def_20 .= ((('not' a) '&' b) 'or' (a '&' ('not' b))) . x by BVFUNC_1:def_4 .= (a 'xor' b) . x by BVFUNC_4:9 ; hence (a 'xor' b) . x = ((a 'or' b) '&' (('not' a) 'or' ('not' b))) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:14 for Y being non empty set for a being Function of Y,BOOLEAN holds a 'xor' (I_el Y) = 'not' a proof let Y be non empty set ; ::_thesis: for a being Function of Y,BOOLEAN holds a 'xor' (I_el Y) = 'not' a let a be Function of Y,BOOLEAN; ::_thesis: a 'xor' (I_el Y) = 'not' a let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'xor' (I_el Y)) . x = ('not' a) . x (a 'xor' (I_el Y)) . x = ((('not' a) '&' (I_el Y)) 'or' (a '&' ('not' (I_el Y)))) . x by BVFUNC_4:9 .= ((('not' a) '&' (I_el Y)) 'or' (a '&' (O_el Y))) . x by BVFUNC_1:2 .= ((('not' a) '&' (I_el Y)) 'or' (O_el Y)) . x by BVFUNC_1:5 .= (('not' a) '&' (I_el Y)) . x by BVFUNC_1:9 .= ('not' a) . x by BVFUNC_1:6 ; hence (a 'xor' (I_el Y)) . x = ('not' a) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:15 for Y being non empty set for a being Function of Y,BOOLEAN holds a 'xor' (O_el Y) = a proof let Y be non empty set ; ::_thesis: for a being Function of Y,BOOLEAN holds a 'xor' (O_el Y) = a let a be Function of Y,BOOLEAN; ::_thesis: a 'xor' (O_el Y) = a let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'xor' (O_el Y)) . x = a . x (a 'xor' (O_el Y)) . x = ((('not' a) '&' (O_el Y)) 'or' (a '&' ('not' (O_el Y)))) . x by BVFUNC_4:9 .= ((('not' a) '&' (O_el Y)) 'or' (a '&' (I_el Y))) . x by BVFUNC_1:2 .= ((('not' a) '&' (O_el Y)) 'or' a) . x by BVFUNC_1:6 .= ((O_el Y) 'or' a) . x by BVFUNC_1:5 .= a . x by BVFUNC_1:9 ; hence (a 'xor' (O_el Y)) . x = a . x ; ::_thesis: verum end; theorem :: BVFUNC_8:16 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' b = ('not' a) 'xor' ('not' b) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a 'xor' b = ('not' a) 'xor' ('not' b) let a, b be Function of Y,BOOLEAN; ::_thesis: a 'xor' b = ('not' a) 'xor' ('not' b) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'xor' b) . x = (('not' a) 'xor' ('not' b)) . x (('not' a) 'xor' ('not' b)) . x = ((('not' ('not' a)) '&' ('not' b)) 'or' (('not' a) '&' ('not' ('not' b)))) . x by BVFUNC_4:9 .= (a 'xor' b) . x by BVFUNC_4:9 ; hence (a 'xor' b) . x = (('not' a) 'xor' ('not' b)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:17 for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a 'xor' b) = a 'xor' ('not' b) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds 'not' (a 'xor' b) = a 'xor' ('not' b) let a, b be Function of Y,BOOLEAN; ::_thesis: 'not' (a 'xor' b) = a 'xor' ('not' b) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: ('not' (a 'xor' b)) . x = (a 'xor' ('not' b)) . x ('not' (a 'xor' b)) . x = ('not' ((('not' a) '&' b) 'or' (a '&' ('not' b)))) . x by BVFUNC_4:9 .= (('not' (('not' a) '&' b)) '&' ('not' (a '&' ('not' b)))) . x by BVFUNC_1:13 .= ((('not' ('not' a)) 'or' ('not' b)) '&' ('not' (a '&' ('not' b)))) . x by BVFUNC_1:14 .= ((a 'or' ('not' b)) '&' (('not' a) 'or' ('not' ('not' b)))) . x by BVFUNC_1:14 .= (((a 'or' ('not' b)) '&' ('not' a)) 'or' ((a 'or' ('not' b)) '&' b)) . x by BVFUNC_1:12 .= (((a '&' ('not' a)) 'or' (('not' b) '&' ('not' a))) 'or' ((a 'or' ('not' b)) '&' b)) . x by BVFUNC_1:12 .= (((O_el Y) 'or' (('not' b) '&' ('not' a))) 'or' ((a 'or' ('not' b)) '&' b)) . x by BVFUNC_4:5 .= ((('not' b) '&' ('not' a)) 'or' ((a 'or' ('not' b)) '&' b)) . x by BVFUNC_1:9 .= ((('not' b) '&' ('not' a)) 'or' ((a '&' b) 'or' (('not' b) '&' b))) . x by BVFUNC_1:12 .= ((('not' b) '&' ('not' a)) 'or' ((a '&' b) 'or' (O_el Y))) . x by BVFUNC_4:5 .= ((('not' a) '&' ('not' b)) 'or' (a '&' ('not' ('not' b)))) . x by BVFUNC_1:9 .= (a 'xor' ('not' b)) . x by BVFUNC_4:9 ; hence ('not' (a 'xor' b)) . x = (a 'xor' ('not' b)) . x ; ::_thesis: verum end; theorem Th18: :: BVFUNC_8:18 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a 'or' ('not' b)) '&' (('not' a) 'or' b) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a 'or' ('not' b)) '&' (('not' a) 'or' b) let a, b be Function of Y,BOOLEAN; ::_thesis: a 'eqv' b = (a 'or' ('not' b)) '&' (('not' a) 'or' b) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'eqv' b) . x = ((a 'or' ('not' b)) '&' (('not' a) 'or' b)) . x ((a 'or' ('not' b)) '&' (('not' a) 'or' b)) . x = ((a 'or' ('not' b)) '&' (a 'imp' b)) . x by BVFUNC_4:8 .= ((a 'imp' b) '&' (b 'imp' a)) . x by BVFUNC_4:8 .= (a 'eqv' b) . x by BVFUNC_4:7 ; hence (a 'eqv' b) . x = ((a 'or' ('not' b)) '&' (('not' a) 'or' b)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:19 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a '&' b) 'or' (('not' a) '&' ('not' b)) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a '&' b) 'or' (('not' a) '&' ('not' b)) let a, b be Function of Y,BOOLEAN; ::_thesis: a 'eqv' b = (a '&' b) 'or' (('not' a) '&' ('not' b)) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'eqv' b) . x = ((a '&' b) 'or' (('not' a) '&' ('not' b))) . x ((a '&' b) 'or' (('not' a) '&' ('not' b))) . x = (((a '&' b) 'or' ('not' a)) '&' ((a '&' b) 'or' ('not' b))) . x by BVFUNC_1:11 .= (((a 'or' ('not' a)) '&' (b 'or' ('not' a))) '&' ((a '&' b) 'or' ('not' b))) . x by BVFUNC_1:11 .= (((a 'or' ('not' a)) '&' (b 'or' ('not' a))) '&' ((a 'or' ('not' b)) '&' (b 'or' ('not' b)))) . x by BVFUNC_1:11 .= (((I_el Y) '&' (b 'or' ('not' a))) '&' ((a 'or' ('not' b)) '&' (b 'or' ('not' b)))) . x by BVFUNC_4:6 .= (((I_el Y) '&' (b 'or' ('not' a))) '&' ((a 'or' ('not' b)) '&' (I_el Y))) . x by BVFUNC_4:6 .= ((b 'or' ('not' a)) '&' ((a 'or' ('not' b)) '&' (I_el Y))) . x by BVFUNC_1:6 .= ((b 'or' ('not' a)) '&' (a 'or' ('not' b))) . x by BVFUNC_1:6 .= (a 'eqv' b) . x by Th18 ; hence (a 'eqv' b) . x = ((a '&' b) 'or' (('not' a) '&' ('not' b))) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:20 for Y being non empty set for a being Function of Y,BOOLEAN holds a 'eqv' (I_el Y) = a proof let Y be non empty set ; ::_thesis: for a being Function of Y,BOOLEAN holds a 'eqv' (I_el Y) = a let a be Function of Y,BOOLEAN; ::_thesis: a 'eqv' (I_el Y) = a let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'eqv' (I_el Y)) . x = a . x (a 'eqv' (I_el Y)) . x = ((a 'imp' (I_el Y)) '&' ((I_el Y) 'imp' a)) . x by BVFUNC_4:7 .= ((('not' a) 'or' (I_el Y)) '&' ((I_el Y) 'imp' a)) . x by BVFUNC_4:8 .= ((('not' a) 'or' (I_el Y)) '&' (('not' (I_el Y)) 'or' a)) . x by BVFUNC_4:8 .= ((I_el Y) '&' (('not' (I_el Y)) 'or' a)) . x by BVFUNC_1:10 .= ((I_el Y) '&' ((O_el Y) 'or' a)) . x by BVFUNC_1:2 .= ((I_el Y) '&' a) . x by BVFUNC_1:9 .= a . x by BVFUNC_1:6 ; hence (a 'eqv' (I_el Y)) . x = a . x ; ::_thesis: verum end; theorem :: BVFUNC_8:21 for Y being non empty set for a being Function of Y,BOOLEAN holds a 'eqv' (O_el Y) = 'not' a proof let Y be non empty set ; ::_thesis: for a being Function of Y,BOOLEAN holds a 'eqv' (O_el Y) = 'not' a let a be Function of Y,BOOLEAN; ::_thesis: a 'eqv' (O_el Y) = 'not' a let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: (a 'eqv' (O_el Y)) . x = ('not' a) . x (a 'eqv' (O_el Y)) . x = ((a 'imp' (O_el Y)) '&' ((O_el Y) 'imp' a)) . x by BVFUNC_4:7 .= ((('not' a) 'or' (O_el Y)) '&' ((O_el Y) 'imp' a)) . x by BVFUNC_4:8 .= ((('not' a) 'or' (O_el Y)) '&' (('not' (O_el Y)) 'or' a)) . x by BVFUNC_4:8 .= (('not' a) '&' (('not' (O_el Y)) 'or' a)) . x by BVFUNC_1:9 .= (('not' a) '&' ((I_el Y) 'or' a)) . x by BVFUNC_1:2 .= (('not' a) '&' (I_el Y)) . x by BVFUNC_1:10 .= ('not' a) . x by BVFUNC_1:6 ; hence (a 'eqv' (O_el Y)) . x = ('not' a) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:22 for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a 'eqv' b) = a 'eqv' ('not' b) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds 'not' (a 'eqv' b) = a 'eqv' ('not' b) let a, b be Function of Y,BOOLEAN; ::_thesis: 'not' (a 'eqv' b) = a 'eqv' ('not' b) let x be Element of Y; :: according to FUNCT_2:def_8 ::_thesis: ('not' (a 'eqv' b)) . x = (a 'eqv' ('not' b)) . x ('not' (a 'eqv' b)) . x = ('not' ((a 'imp' b) '&' (b 'imp' a))) . x by BVFUNC_4:7 .= ('not' ((('not' a) 'or' b) '&' (b 'imp' a))) . x by BVFUNC_4:8 .= ('not' ((('not' a) 'or' b) '&' (('not' b) 'or' a))) . x by BVFUNC_4:8 .= (('not' (('not' a) 'or' b)) 'or' ('not' (('not' b) 'or' a))) . x by BVFUNC_1:14 .= ((('not' ('not' a)) '&' ('not' b)) 'or' ('not' (('not' b) 'or' a))) . x by BVFUNC_1:13 .= ((a '&' ('not' b)) 'or' (('not' ('not' b)) '&' ('not' a))) . x by BVFUNC_1:13 .= (((a '&' ('not' b)) 'or' b) '&' ((a '&' ('not' b)) 'or' ('not' a))) . x by BVFUNC_1:11 .= (((a 'or' b) '&' (('not' b) 'or' b)) '&' ((a '&' ('not' b)) 'or' ('not' a))) . x by BVFUNC_1:11 .= (((a 'or' b) '&' (('not' b) 'or' b)) '&' ((a 'or' ('not' a)) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_1:11 .= (((a 'or' b) '&' (I_el Y)) '&' ((a 'or' ('not' a)) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_4:6 .= (((a 'or' b) '&' (I_el Y)) '&' ((I_el Y) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_4:6 .= ((a 'or' b) '&' ((I_el Y) '&' (('not' b) 'or' ('not' a)))) . x by BVFUNC_1:6 .= ((('not' a) 'or' ('not' b)) '&' (('not' ('not' b)) 'or' a)) . x by BVFUNC_1:6 .= ((('not' a) 'or' ('not' b)) '&' (('not' b) 'imp' a)) . x by BVFUNC_4:8 .= ((a 'imp' ('not' b)) '&' (('not' b) 'imp' a)) . x by BVFUNC_4:8 .= (a 'eqv' ('not' b)) . x by BVFUNC_4:7 ; hence ('not' (a 'eqv' b)) . x = (a 'eqv' ('not' b)) . x ; ::_thesis: verum end; theorem :: BVFUNC_8:23 for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' a '<' (a 'imp' b) 'eqv' ('not' a) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds 'not' a '<' (a 'imp' b) 'eqv' ('not' a) let a, b be Function of Y,BOOLEAN; ::_thesis: 'not' a '<' (a 'imp' b) 'eqv' ('not' a) let z be Element of Y; :: according to BVFUNC_1:def_12 ::_thesis: ( not ('not' a) . z = TRUE or ((a 'imp' b) 'eqv' ('not' a)) . z = TRUE ) assume A1: ('not' a) . z = TRUE ; ::_thesis: ((a 'imp' b) 'eqv' ('not' a)) . z = TRUE then 'not' (a . z) = TRUE by MARGREL1:def_19; then A2: a . z = FALSE by MARGREL1:11; ((a 'imp' b) 'eqv' ('not' a)) . z = ((('not' a) 'or' b) 'eqv' ('not' a)) . z by BVFUNC_4:8 .= (((('not' a) 'or' b) 'imp' ('not' a)) '&' (('not' a) 'imp' (('not' a) 'or' b))) . z by BVFUNC_4:7 .= ((('not' (('not' a) 'or' b)) 'or' ('not' a)) '&' (('not' a) 'imp' (('not' a) 'or' b))) . z by BVFUNC_4:8 .= ((('not' (('not' a) 'or' b)) 'or' ('not' a)) '&' (('not' ('not' a)) 'or' (('not' a) 'or' b))) . z by BVFUNC_4:8 .= ((('not' (('not' a) 'or' b)) 'or' ('not' a)) . z) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def_20 .= ((('not' (('not' a) 'or' b)) . z) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by BVFUNC_1:def_4 .= (('not' ((('not' a) 'or' b) . z)) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def_19 .= (('not' ((('not' a) . z) 'or' (b . z))) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by BVFUNC_1:def_4 .= ((('not' ('not' (a . z))) '&' ('not' (b . z))) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def_19 .= (((a . z) '&' ('not' (b . z))) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) . z) 'or' ((('not' a) 'or' b) . z)) by BVFUNC_1:def_4 .= (((a . z) '&' ('not' (b . z))) 'or' (('not' a) . z)) '&' ((a . z) 'or' ((('not' a) . z) 'or' (b . z))) by BVFUNC_1:def_4 .= TRUE '&' (FALSE 'or' (TRUE 'or' (b . z))) by A1, A2, BINARITH:10 .= FALSE 'or' (TRUE 'or' (b . z)) by MARGREL1:14 .= TRUE 'or' (b . z) by BINARITH:3 .= TRUE by BINARITH:10 ; hence ((a 'imp' b) 'eqv' ('not' a)) . z = TRUE ; ::_thesis: verum end; theorem :: BVFUNC_8:24 for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' a '<' (b 'imp' a) 'eqv' ('not' b) proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds 'not' a '<' (b 'imp' a) 'eqv' ('not' b) let a, b be Function of Y,BOOLEAN; ::_thesis: 'not' a '<' (b 'imp' a) 'eqv' ('not' b) let z be Element of Y; :: according to BVFUNC_1:def_12 ::_thesis: ( not ('not' a) . z = TRUE or ((b 'imp' a) 'eqv' ('not' b)) . z = TRUE ) assume ('not' a) . z = TRUE ; ::_thesis: ((b 'imp' a) 'eqv' ('not' b)) . z = TRUE then A1: 'not' (a . z) = TRUE by MARGREL1:def_19; ((b 'imp' a) 'eqv' ('not' b)) . z = ((('not' b) 'or' a) 'eqv' ('not' b)) . z by BVFUNC_4:8 .= (((('not' b) 'or' a) 'imp' ('not' b)) '&' (('not' b) 'imp' (('not' b) 'or' a))) . z by BVFUNC_4:7 .= ((('not' (('not' b) 'or' a)) 'or' ('not' b)) '&' (('not' b) 'imp' (('not' b) 'or' a))) . z by BVFUNC_4:8 .= ((('not' (('not' b) 'or' a)) 'or' ('not' b)) '&' (('not' ('not' b)) 'or' (('not' b) 'or' a))) . z by BVFUNC_4:8 .= ((('not' (('not' b) 'or' a)) 'or' ('not' b)) . z) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by MARGREL1:def_20 .= ((('not' (('not' b) 'or' a)) . z) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by BVFUNC_1:def_4 .= (('not' ((('not' b) 'or' a) . z)) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by MARGREL1:def_19 .= (('not' ((('not' b) . z) 'or' (a . z))) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by BVFUNC_1:def_4 .= ((('not' ('not' (b . z))) '&' ('not' (a . z))) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by MARGREL1:def_19 .= (((b . z) '&' ('not' (a . z))) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) . z) 'or' ((('not' b) 'or' a) . z)) by BVFUNC_1:def_4 .= (((b . z) '&' ('not' (a . z))) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) . z) 'or' ((('not' b) . z) 'or' (a . z))) by BVFUNC_1:def_4 .= (((b . z) '&' ('not' (a . z))) 'or' (('not' b) . z)) '&' ((b . z) 'or' (('not' (b . z)) 'or' (a . z))) by MARGREL1:def_19 .= ((TRUE '&' (b . z)) 'or' (('not' b) . z)) '&' ((b . z) 'or' (('not' (b . z)) 'or' FALSE)) by A1, MARGREL1:11 .= ((b . z) 'or' (('not' b) . z)) '&' ((b . z) 'or' (('not' (b . z)) 'or' FALSE)) by MARGREL1:14 .= ((b . z) 'or' ('not' (b . z))) '&' ((b . z) 'or' (('not' (b . z)) 'or' FALSE)) by MARGREL1:def_19 .= TRUE '&' ((b . z) 'or' (('not' (b . z)) 'or' FALSE)) by XBOOLEAN:102 .= (b . z) 'or' (('not' (b . z)) 'or' FALSE) by MARGREL1:14 .= ((b . z) 'or' ('not' (b . z))) 'or' FALSE by BINARITH:11 .= TRUE 'or' FALSE by XBOOLEAN:102 .= TRUE by BINARITH:10 ; hence ((b 'imp' a) 'eqv' ('not' b)) . z = TRUE ; ::_thesis: verum end; theorem :: BVFUNC_8:25 for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '<' ((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds a '<' ((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a let a, b be Function of Y,BOOLEAN; ::_thesis: a '<' ((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a let z be Element of Y; :: according to BVFUNC_1:def_12 ::_thesis: ( not a . z = TRUE or (((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z = TRUE ) assume A1: a . z = TRUE ; ::_thesis: (((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z = TRUE A2: ((a 'or' b) 'eqv' (b 'or' a)) . z = (((a 'or' b) 'imp' (a 'or' b)) '&' ((a 'or' b) 'imp' (a 'or' b))) . z by BVFUNC_4:7 .= (('not' (a 'or' b)) 'or' (a 'or' b)) . z by BVFUNC_4:8 .= (I_el Y) . z by BVFUNC_4:6 .= TRUE by BVFUNC_1:def_11 ; (((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z = ((((a 'or' b) 'eqv' (a 'or' b)) 'imp' a) '&' (a 'imp' ((a 'or' b) 'eqv' (a 'or' b)))) . z by BVFUNC_4:7 .= ((((a 'or' b) 'eqv' (a 'or' b)) 'imp' a) . z) '&' ((a 'imp' ((a 'or' b) 'eqv' (a 'or' b))) . z) by MARGREL1:def_20 .= ((('not' ((a 'or' b) 'eqv' (a 'or' b))) 'or' a) . z) '&' ((a 'imp' ((a 'or' b) 'eqv' (a 'or' b))) . z) by BVFUNC_4:8 .= ((('not' ((a 'or' b) 'eqv' (a 'or' b))) 'or' a) . z) '&' ((('not' a) 'or' ((a 'or' b) 'eqv' (a 'or' b))) . z) by BVFUNC_4:8 .= ((('not' ((a 'or' b) 'eqv' (a 'or' b))) . z) 'or' (a . z)) '&' ((('not' a) 'or' ((a 'or' b) 'eqv' (a 'or' b))) . z) by BVFUNC_1:def_4 .= ((('not' ((a 'or' b) 'eqv' (a 'or' b))) . z) 'or' (a . z)) '&' ((('not' a) . z) 'or' (((a 'or' b) 'eqv' (a 'or' b)) . z)) by BVFUNC_1:def_4 .= (('not' (((a 'or' b) 'eqv' (a 'or' b)) . z)) 'or' (a . z)) '&' ((('not' a) . z) 'or' (((a 'or' b) 'eqv' (a 'or' b)) . z)) by MARGREL1:def_19 .= (FALSE 'or' (a . z)) '&' ((('not' a) . z) 'or' TRUE) by A2, MARGREL1:11 .= (a . z) '&' ((('not' a) . z) 'or' TRUE) by BINARITH:3 .= (a . z) '&' TRUE by BINARITH:10 .= TRUE by A1 ; hence (((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z = TRUE ; ::_thesis: verum end; theorem :: BVFUNC_8:26 for Y being non empty set for a being Function of Y,BOOLEAN holds a 'imp' (('not' a) 'eqv' ('not' a)) = I_el Y proof let Y be non empty set ; ::_thesis: for a being Function of Y,BOOLEAN holds a 'imp' (('not' a) 'eqv' ('not' a)) = I_el Y let a be Function of Y,BOOLEAN; ::_thesis: a 'imp' (('not' a) 'eqv' ('not' a)) = I_el Y for x being Element of Y holds (a 'imp' (('not' a) 'eqv' ('not' a))) . x = TRUE proof let x be Element of Y; ::_thesis: (a 'imp' (('not' a) 'eqv' ('not' a))) . x = TRUE (a 'imp' (('not' a) 'eqv' ('not' a))) . x = (('not' a) 'or' (('not' a) 'eqv' ('not' a))) . x by BVFUNC_4:8 .= (('not' a) 'or' ((('not' a) 'imp' ('not' a)) '&' (('not' a) 'imp' ('not' a)))) . x by BVFUNC_4:7 .= (('not' a) 'or' (('not' ('not' a)) 'or' ('not' a))) . x by BVFUNC_4:8 .= (('not' a) 'or' (I_el Y)) . x by BVFUNC_4:6 .= TRUE by BVFUNC_1:def_11, BVFUNC_1:10 ; hence (a 'imp' (('not' a) 'eqv' ('not' a))) . x = TRUE ; ::_thesis: verum end; hence a 'imp' (('not' a) 'eqv' ('not' a)) = I_el Y by BVFUNC_1:def_11; ::_thesis: verum end; theorem :: BVFUNC_8:27 for Y being non empty set for a, b being Function of Y,BOOLEAN holds ((a 'imp' b) 'imp' a) 'imp' a = I_el Y proof let Y be non empty set ; ::_thesis: for a, b being Function of Y,BOOLEAN holds ((a 'imp' b) 'imp' a) 'imp' a = I_el Y let a, b be Function of Y,BOOLEAN; ::_thesis: ((a 'imp' b) 'imp' a) 'imp' a = I_el Y for x being Element of Y holds (((a 'imp' b) 'imp' a) 'imp' a) . x = TRUE proof let x be Element of Y; ::_thesis: (((a 'imp' b) 'imp' a) 'imp' a) . x = TRUE (((a 'imp' b) 'imp' a) 'imp' a) . x = (('not' ((a 'imp' b) 'imp' a)) 'or' a) . x by BVFUNC_4:8 .= (('not' (('not' (a 'imp' b)) 'or' a)) 'or' a) . x by BVFUNC_4:8 .= (('not' (('not' (('not' a) 'or' b)) 'or' a)) 'or' a) . x by BVFUNC_4:8 .= (('not' ((('not' ('not' a)) '&' ('not' b)) 'or' a)) 'or' a) . x by BVFUNC_1:13 .= (('not' ((a 'or' a) '&' (('not' b) 'or' a))) 'or' a) . x by BVFUNC_1:11 .= ((('not' a) 'or' ('not' (('not' b) 'or' a))) 'or' a) . x by BVFUNC_1:14 .= ((('not' a) 'or' (('not' ('not' b)) '&' ('not' a))) 'or' a) . x by BVFUNC_1:13 .= (((('not' a) 'or' b) '&' (('not' a) 'or' ('not' a))) 'or' a) . x by BVFUNC_1:11 .= (((('not' a) 'or' b) 'or' a) '&' (('not' a) 'or' a)) . x by BVFUNC_1:11 .= (((('not' a) 'or' b) 'or' a) '&' (I_el Y)) . x by BVFUNC_4:6 .= ((('not' a) 'or' b) 'or' a) . x by BVFUNC_1:6 .= (b 'or' (('not' a) 'or' a)) . x by BVFUNC_1:8 .= (b 'or' (I_el Y)) . x by BVFUNC_4:6 .= TRUE by BVFUNC_1:def_11, BVFUNC_1:10 ; hence (((a 'imp' b) 'imp' a) 'imp' a) . x = TRUE ; ::_thesis: verum end; hence ((a 'imp' b) 'imp' a) 'imp' a = I_el Y by BVFUNC_1:def_11; ::_thesis: verum end; theorem :: BVFUNC_8:28 for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = I_el Y proof let Y be non empty set ; ::_thesis: for a, b, c, d being Function of Y,BOOLEAN holds (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = I_el Y let a, b, c, d be Function of Y,BOOLEAN; ::_thesis: (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = I_el Y for x being Element of Y holds ((((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b))) . x = TRUE proof let x be Element of Y; ::_thesis: ((((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b))) . x = TRUE (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = ('not' (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_4:8 .= ('not' (((('not' a) 'or' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_4:8 .= ('not' (((('not' a) 'or' c) '&' (('not' b) 'or' d)) '&' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_4:8 .= (('not' ((('not' a) 'or' c) '&' (('not' b) 'or' d))) 'or' ('not' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:14 .= ((('not' (('not' a) 'or' c)) 'or' ('not' (('not' b) 'or' d))) 'or' ('not' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:14 .= (((('not' ('not' a)) '&' ('not' c)) 'or' ('not' (('not' b) 'or' d))) 'or' ('not' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:13 .= (((a '&' ('not' c)) 'or' (('not' ('not' b)) '&' ('not' d))) 'or' ('not' (('not' c) 'or' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:13 .= (((a '&' ('not' c)) 'or' (b '&' ('not' d))) 'or' (('not' ('not' c)) '&' ('not' ('not' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:13 .= ((a '&' ('not' c)) 'or' ((b '&' ('not' d)) 'or' (c '&' d))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:8 .= ((a '&' ('not' c)) 'or' ((b 'or' (c '&' d)) '&' (('not' d) 'or' (c '&' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:11 .= ((a '&' ('not' c)) 'or' ((b 'or' (c '&' d)) '&' ((('not' d) 'or' c) '&' (('not' d) 'or' d)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:11 .= ((a '&' ('not' c)) 'or' ((b 'or' (c '&' d)) '&' ((('not' d) 'or' c) '&' (I_el Y)))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_4:6 .= ((a '&' ('not' c)) 'or' ((b 'or' (c '&' d)) '&' (('not' d) 'or' c))) 'or' (('not' a) 'or' ('not' b)) by BVFUNC_1:6 .= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' ((a '&' ('not' c)) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:8 .= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' ((a 'or' (('not' a) 'or' ('not' b))) '&' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:11 .= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' (((a 'or' ('not' a)) 'or' ('not' b)) '&' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:8 .= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' (((I_el Y) 'or' ('not' b)) '&' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_4:6 .= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' ((I_el Y) '&' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:10 .= ((b 'or' (c '&' d)) '&' (('not' d) 'or' c)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:6 .= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' ((('not' d) 'or' c) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:11 .= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' (((('not' d) 'or' c) 'or' ('not' c)) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:8 .= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' ((('not' d) 'or' (c 'or' ('not' c))) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:8 .= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' ((('not' d) 'or' (I_el Y)) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_4:6 .= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' ((I_el Y) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:10 .= ((b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) '&' (I_el Y) by BVFUNC_1:10 .= (b 'or' (c '&' d)) 'or' (('not' c) 'or' (('not' a) 'or' ('not' b))) by BVFUNC_1:6 .= (c '&' d) 'or' (b 'or' (('not' c) 'or' (('not' a) 'or' ('not' b)))) by BVFUNC_1:8 .= (c '&' d) 'or' ((b 'or' (('not' b) 'or' ('not' a))) 'or' ('not' c)) by BVFUNC_1:8 .= (c '&' d) 'or' (((b 'or' ('not' b)) 'or' ('not' a)) 'or' ('not' c)) by BVFUNC_1:8 .= (c '&' d) 'or' (((I_el Y) 'or' ('not' a)) 'or' ('not' c)) by BVFUNC_4:6 .= (c '&' d) 'or' ((I_el Y) 'or' ('not' c)) by BVFUNC_1:10 .= (c '&' d) 'or' (I_el Y) by BVFUNC_1:10 .= I_el Y by BVFUNC_1:10 ; hence ((((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b))) . x = TRUE by BVFUNC_1:def_11; ::_thesis: verum end; hence (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = I_el Y by BVFUNC_1:def_11; ::_thesis: verum end; theorem :: BVFUNC_8:29 for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) = I_el Y proof let Y be non empty set ; ::_thesis: for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((a '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' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) = I_el Y for x being Element of Y holds ((a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))) . x = TRUE proof let x be Element of Y; ::_thesis: ((a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))) . x = TRUE (a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) = ('not' (a 'imp' b)) 'or' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8 .= ('not' (('not' a) 'or' b)) 'or' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8 .= ('not' (('not' a) 'or' b)) 'or' ((('not' a) 'or' (b 'imp' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8 .= ('not' (('not' a) 'or' b)) 'or' ((('not' a) 'or' (('not' b) 'or' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8 .= ('not' (('not' a) 'or' b)) 'or' ((('not' a) 'or' (('not' b) 'or' c)) 'imp' (('not' a) 'or' c)) by BVFUNC_4:8 .= ('not' (('not' a) 'or' b)) 'or' (('not' (('not' a) 'or' (('not' b) 'or' c))) 'or' (('not' a) 'or' c)) by BVFUNC_4:8 .= (('not' ('not' a)) '&' ('not' b)) 'or' (('not' (('not' a) 'or' (('not' b) 'or' c))) 'or' (('not' a) 'or' c)) by BVFUNC_1:13 .= (('not' ('not' a)) '&' ('not' b)) 'or' ((('not' ('not' a)) '&' ('not' (('not' b) 'or' c))) 'or' (('not' a) 'or' c)) by BVFUNC_1:13 .= (a '&' ('not' b)) 'or' ((('not' ('not' a)) '&' (('not' ('not' b)) '&' ('not' c))) 'or' (('not' a) 'or' c)) by BVFUNC_1:13 .= (a '&' ('not' b)) 'or' ((a 'or' (('not' a) 'or' c)) '&' ((b '&' ('not' c)) 'or' (('not' a) 'or' c))) by BVFUNC_1:11 .= (a '&' ('not' b)) 'or' (((a 'or' ('not' a)) 'or' c) '&' ((b '&' ('not' c)) 'or' (('not' a) 'or' c))) by BVFUNC_1:8 .= (a '&' ('not' b)) 'or' (((I_el Y) 'or' c) '&' ((b '&' ('not' c)) 'or' (('not' a) 'or' c))) by BVFUNC_4:6 .= (a '&' ('not' b)) 'or' ((I_el Y) '&' ((b '&' ('not' c)) 'or' (('not' a) 'or' c))) by BVFUNC_1:10 .= (a '&' ('not' b)) 'or' ((b '&' ('not' c)) 'or' (('not' a) 'or' c)) by BVFUNC_1:6 .= (a '&' ('not' b)) 'or' ((b 'or' (('not' a) 'or' c)) '&' (('not' c) 'or' (('not' a) 'or' c))) by BVFUNC_1:11 .= (a '&' ('not' b)) 'or' ((b 'or' (('not' a) 'or' c)) '&' ((('not' c) 'or' c) 'or' ('not' a))) by BVFUNC_1:8 .= (a '&' ('not' b)) 'or' ((b 'or' (('not' a) 'or' c)) '&' ((I_el Y) 'or' ('not' a))) by BVFUNC_4:6 .= (a '&' ('not' b)) 'or' ((b 'or' (('not' a) 'or' c)) '&' (I_el Y)) by BVFUNC_1:10 .= (a '&' ('not' b)) 'or' (b 'or' (('not' a) 'or' c)) by BVFUNC_1:6 .= (a 'or' (b 'or' (('not' a) 'or' c))) '&' (('not' b) 'or' (b 'or' (('not' a) 'or' c))) by BVFUNC_1:11 .= (a 'or' (b 'or' (('not' a) 'or' c))) '&' ((('not' b) 'or' b) 'or' (('not' a) 'or' c)) by BVFUNC_1:8 .= (a 'or' (b 'or' (('not' a) 'or' c))) '&' ((I_el Y) 'or' (('not' a) 'or' c)) by BVFUNC_4:6 .= (a 'or' (b 'or' (('not' a) 'or' c))) '&' (I_el Y) by BVFUNC_1:10 .= a 'or' (b 'or' (('not' a) 'or' c)) by BVFUNC_1:6 .= a 'or' ((('not' a) 'or' b) 'or' c) by BVFUNC_1:8 .= (a 'or' (('not' a) 'or' b)) 'or' c by BVFUNC_1:8 .= ((a 'or' ('not' a)) 'or' b) 'or' c by BVFUNC_1:8 .= ((I_el Y) 'or' b) 'or' c by BVFUNC_4:6 .= (I_el Y) 'or' c by BVFUNC_1:10 .= I_el Y by BVFUNC_1:10 ; hence ((a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))) . x = TRUE by BVFUNC_1:def_11; ::_thesis: verum end; hence (a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) = I_el Y by BVFUNC_1:def_11; ::_thesis: verum end;