let p be Element of CQC-WFF ; :: thesis: ( ( for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = p & S `2 = Sub ) ) implies for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = 'not' p & S `2 = Sub ) )
assume A1: for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = p & S `2 = Sub ) ; :: thesis: for Sub being CQC_Substitution ex S being Element of CQC-Sub-WFF st
( S `1 = 'not' p & S `2 = Sub )
let Sub be CQC_Substitution; :: thesis: ex S being Element of CQC-Sub-WFF st
( S `1 = 'not' p & S `2 = Sub )
consider S being Element of CQC-Sub-WFF such that
A2: ( S `1 = p & S `2 = Sub ) by A1;
S = [p,Sub] by A2, SUBSTUT1:10;
then [p,Sub] in QC-Sub-WFF ;
then [(@ p),Sub] in QC-Sub-WFF by QC_LANG1:def 12;
then [(<*[1,0]*> ^ (@ p)),Sub] in QC-Sub-WFF by SUBSTUT1:def 16;
then reconsider S = [('not' p),Sub] as Element of QC-Sub-WFF by QC_LANG1:def 14;
take S ; :: thesis: ( S is Element of CQC-Sub-WFF & S `1 = 'not' p & S `2 = Sub )
S `1 = 'not' p by MCART_1:7;
then S in CQC-Sub-WFF by SUBSTUT1:def 39;
then reconsider S = S as Element of CQC-Sub-WFF ;
S `2 = Sub by MCART_1:7;
hence ( S is Element of CQC-Sub-WFF & S `1 = 'not' p & S `2 = Sub ) by MCART_1:7; :: thesis: verum