Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Propositional Calculus for Boolean Valued Functions. Part III

**by****Shunichi Kobayashi**- Received April 23, 1999
- MML identifier: BVFUNC_7

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_2, MARGREL1, BVFUNC_1, ZF_LANG, FUNCT_1, RELAT_1, PARTIT1; notation TARSKI, XBOOLE_0, SUBSET_1, MARGREL1, VALUAT_1, RELAT_1, FUNCT_1, FRAENKEL, BINARITH, BVFUNC_1; constructors BINARITH, BVFUNC_1; clusters MARGREL1, VALUAT_1, BINARITH, FRAENKEL; requirements SUBSET, BOOLE; begin ::Chap. 1 Propositional Calculus reserve Y for non empty set; theorem :: BVFUNC_7:1 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b) '&' ('not' a 'imp' b) = b; theorem :: BVFUNC_7:2 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b) '&' (a 'imp' 'not' b) = 'not' a; theorem :: BVFUNC_7:3 for a,b,c being Element of Funcs(Y,BOOLEAN) holds a 'imp' (b 'or' c) = (a 'imp' b) 'or' (a 'imp' c); theorem :: BVFUNC_7:4 for a,b,c being Element of Funcs(Y,BOOLEAN) holds a 'imp' (b '&' c) = (a 'imp' b) '&' (a 'imp' c); theorem :: BVFUNC_7:5 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'or' b) 'imp' c = (a 'imp' c) '&' (b 'imp' c); theorem :: BVFUNC_7:6 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a '&' b) 'imp' c = (a 'imp' c) 'or' (b 'imp' c); theorem :: BVFUNC_7:7 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a '&' b) 'imp' c = a 'imp' (b 'imp' c); theorem :: BVFUNC_7:8 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a '&' b) 'imp' c = a 'imp' ('not' b 'or' c); theorem :: BVFUNC_7:9 for a,b,c being Element of Funcs(Y,BOOLEAN) holds a 'imp' (b 'or' c) = (a '&' 'not' b) 'imp' c; theorem :: BVFUNC_7:10 for a,b being Element of Funcs(Y,BOOLEAN) holds a '&' (a 'imp' b) = a '&' b; theorem :: BVFUNC_7:11 for a,b being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b) '&' 'not' b = 'not' a '&' 'not' b; theorem :: BVFUNC_7:12 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b) '&' (b 'imp' c) = (a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c); theorem :: BVFUNC_7:13 for a being Element of Funcs(Y,BOOLEAN) holds I_el(Y) 'imp' a = a; theorem :: BVFUNC_7:14 for a being Element of Funcs(Y,BOOLEAN) holds a 'imp' O_el(Y) = 'not' a; theorem :: BVFUNC_7:15 for a being Element of Funcs(Y,BOOLEAN) holds O_el(Y) 'imp' a = I_el(Y); theorem :: BVFUNC_7:16 for a being Element of Funcs(Y,BOOLEAN) holds a 'imp' I_el(Y) = I_el(Y); theorem :: BVFUNC_7:17 for a being Element of Funcs(Y,BOOLEAN) holds a 'imp' 'not' a = 'not' a; theorem :: BVFUNC_7:18 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'imp' b) '<' (c 'imp' a) 'imp' (c 'imp' b); theorem :: BVFUNC_7:19 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'eqv' b) '<' (a 'eqv' c) 'eqv' (b 'eqv' c); theorem :: BVFUNC_7:20 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'eqv' b) '<' (a 'imp' c) 'eqv' (b 'imp' c); theorem :: BVFUNC_7:21 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'eqv' b) '<' (c 'imp' a) 'eqv' (c 'imp' b); theorem :: BVFUNC_7:22 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'eqv' b) '<' (a '&' c) 'eqv' (b '&' c); theorem :: BVFUNC_7:23 for a,b,c being Element of Funcs(Y,BOOLEAN) holds (a 'eqv' b) '<' (a 'or' c) 'eqv' (b 'or' c); theorem :: BVFUNC_7:24 for a,b being Element of Funcs(Y,BOOLEAN) holds a '<' (a 'eqv' b) 'eqv' (b 'eqv' a) 'eqv' a; theorem :: BVFUNC_7:25 for a,b being Element of Funcs(Y,BOOLEAN) holds a '<' (a 'imp' b) 'eqv' b; theorem :: BVFUNC_7:26 for a,b being Element of Funcs(Y,BOOLEAN) holds a '<' (b 'imp' a) 'eqv' a; theorem :: BVFUNC_7:27 for a,b being Element of Funcs(Y,BOOLEAN) holds a '<' (a '&' b) 'eqv' (b '&' a) 'eqv' a;

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