environ vocabulary FUNCT_2, MARGREL1, BVFUNC_1, ZF_LANG, BINARITH; notation XBOOLE_0, SUBSET_1, FRAENKEL, MARGREL1, VALUAT_1, BINARITH, BVFUNC_1; constructors BINARITH, BVFUNC_1; clusters MARGREL1, VALUAT_1, BINARITH, FRAENKEL; begin ::Chap. 1 Propositional Calculus reserve Y for non empty set; theorem :: BVFUNC_6:1 for a,b being Element of Funcs(Y,BOOLEAN) holds a 'imp' (b 'imp' (a '&' b))=I_el(Y); theorem :: BVFUNC_6:2 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))=I_el(Y); theorem :: BVFUNC_6:3 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'or' b) 'eqv' (b 'or' a)=I_el(Y); theorem :: BVFUNC_6:4 for a,b,c being Element of Funcs(Y,BOOLEAN) holds ((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))=I_el(Y); theorem :: BVFUNC_6:5 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)=I_el(Y); theorem :: BVFUNC_6:6 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b)))=I_el(Y); theorem :: BVFUNC_6:7 for a,b,c being Element of Funcs(Y,BOOLEAN) holds ((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))=I_el(Y); theorem :: BVFUNC_6:8 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))=I_el(Y); theorem :: BVFUNC_6:9 for a,b,c being Element of Funcs(Y,BOOLEAN) holds ((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)=I_el(Y); theorem :: BVFUNC_6:10 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'imp' (b '&' 'not' b)) 'imp' 'not' a=I_el(Y); theorem :: BVFUNC_6:11 for a,b,c being Element of Funcs(Y,BOOLEAN) holds ((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))=I_el(Y); theorem :: BVFUNC_6:12 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))=I_el(Y); theorem :: BVFUNC_6:13 for a,b,c being Element of Funcs(Y,BOOLEAN) holds ((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)=I_el(Y); theorem :: BVFUNC_6:14 for a,b,c being Element of Funcs(Y,BOOLEAN) holds ((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))=I_el(Y); theorem :: BVFUNC_6:15 for a,b being Element of Funcs(Y,BOOLEAN) holds (a '&' b)=I_el(Y) implies (a 'or' b)=I_el(Y); theorem :: BVFUNC_6:16 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b)=I_el(Y) implies (a 'or' c) 'imp' (b 'or' c)=I_el(Y); theorem :: BVFUNC_6:17 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b)=I_el(Y) implies (a '&' c) 'imp' (b '&' c)=I_el(Y); theorem :: BVFUNC_6:18 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (c 'imp' a)=I_el(Y) & (c 'imp' b)=I_el(Y) implies c 'imp' (a '&' b)=I_el(Y); theorem :: BVFUNC_6:19 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'imp' c)=I_el(Y) & (b 'imp' c)=I_el(Y) implies (a 'or' b) 'imp' c = I_el(Y); theorem :: BVFUNC_6:20 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'or' b)=I_el(Y) & 'not' a=I_el(Y) implies b=I_el(Y); theorem :: BVFUNC_6:21 for a,b,c,d being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b)=I_el(Y) & (c 'imp' d)=I_el(Y) implies (a '&' c) 'imp' (b '&' d)=I_el(Y); theorem :: BVFUNC_6:22 for a,b,c,d being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b)=I_el(Y) & (c 'imp' d)=I_el(Y) implies (a 'or' c) 'imp' (b 'or' d) =I_el(Y); theorem :: BVFUNC_6:23 for a,b being Element of Funcs(Y,BOOLEAN) holds (a '&' 'not' b) 'imp' 'not' a=I_el(Y) implies (a 'imp' b)=I_el(Y); canceled; theorem :: BVFUNC_6:25 for a,b being Element of Funcs(Y,BOOLEAN) holds a 'imp' 'not' b=I_el(Y) implies b 'imp' 'not' a=I_el(Y); theorem :: BVFUNC_6:26 for a,b being Element of Funcs(Y,BOOLEAN) holds 'not' a 'imp' b=I_el(Y) implies 'not' b 'imp' a=I_el(Y); theorem :: BVFUNC_6:27 for a,b being Element of Funcs(Y,BOOLEAN) holds a 'imp' (a 'or' b)=I_el(Y); theorem :: BVFUNC_6:28 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'or' b) 'imp' ('not' a 'imp' b)=I_el(Y); theorem :: BVFUNC_6:29 for a,b being Element of Funcs(Y,BOOLEAN) holds 'not'( a 'or' b) 'imp' ('not' a '&' 'not' b)=I_el(Y); theorem :: BVFUNC_6:30 for a,b being Element of Funcs(Y,BOOLEAN) holds ('not' a '&' 'not' b) 'imp' 'not'( a 'or' b)=I_el(Y); theorem :: BVFUNC_6:31 for a,b being Element of Funcs(Y,BOOLEAN) holds 'not'( a 'or' b) 'imp' 'not' a=I_el(Y); theorem :: BVFUNC_6:32 for a being Element of Funcs(Y,BOOLEAN) holds (a 'or' a) 'imp' a=I_el(Y); theorem :: BVFUNC_6:33 for a,b being Element of Funcs(Y,BOOLEAN) holds (a '&' 'not' a) 'imp' b=I_el(Y); theorem :: BVFUNC_6:34 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b) 'imp' ('not' a 'or' b)=I_el(Y); theorem :: BVFUNC_6:35 for a,b being Element of Funcs(Y,BOOLEAN) holds (a '&' b) 'imp' 'not'( a 'imp' 'not' b)=I_el(Y); theorem :: BVFUNC_6:36 for a,b being Element of Funcs(Y,BOOLEAN) holds 'not'( a 'imp' 'not' b) 'imp' (a '&' b)=I_el(Y); theorem :: BVFUNC_6:37 for a,b being Element of Funcs(Y,BOOLEAN) holds 'not'( a '&' b) 'imp' ('not' a 'or' 'not' b)=I_el(Y); theorem :: BVFUNC_6:38 for a,b being Element of Funcs(Y,BOOLEAN) holds ('not' a 'or' 'not' b) 'imp' 'not'( a '&' b)=I_el(Y); theorem :: BVFUNC_6:39 for a,b being Element of Funcs(Y,BOOLEAN) holds (a '&' b) 'imp' a=I_el(Y); theorem :: BVFUNC_6:40 for a,b being Element of Funcs(Y,BOOLEAN) holds (a '&' b) 'imp' (a 'or' b)=I_el(Y); theorem :: BVFUNC_6:41 for a,b being Element of Funcs(Y,BOOLEAN) holds (a '&' b) 'imp' b=I_el(Y); theorem :: BVFUNC_6:42 for a being Element of Funcs(Y,BOOLEAN) holds a 'imp' a '&' a=I_el(Y); theorem :: BVFUNC_6:43 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'eqv' b) 'imp' (a 'imp' b)=I_el(Y); theorem :: BVFUNC_6:44 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'eqv' b) 'imp' (b 'imp' a)=I_el(Y); theorem :: BVFUNC_6:45 for a,b,c being Element of Funcs(Y,BOOLEAN) holds ((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))=I_el(Y); theorem :: BVFUNC_6:46 for a,b,c being Element of Funcs(Y,BOOLEAN) holds ((a '&' b) '&' c) 'imp' (a '&' (b '&' c))=I_el(Y); theorem :: BVFUNC_6:47 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)=I_el(Y);