let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable holds
( (Ex x,(p => q)) => ((All x,p) => (Ex x,q)) is valid & ((All x,p) => (Ex x,q)) => (Ex x,(p => q)) is valid )
let x be bound_QC-variable; :: thesis: ( (Ex x,(p => q)) => ((All x,p) => (Ex x,q)) is valid & ((All x,p) => (Ex x,q)) => (Ex x,(p => q)) is valid )
(All x,p) => p is valid by CQC_THE1:105;
then A1: (p => q) => ((All x,p) => q) is valid by LUKASI_1:42;
( not x in still_not-bound_in (All x,p) & not x in still_not-bound_in (Ex x,q) ) by Th5, Th6;
then A2: not x in still_not-bound_in ((All x,p) => (Ex x,q)) by Th7;
q => (Ex x,q) is valid by Th18;
then (p => q) => ((All x,p) => (Ex x,q)) is valid by A1, Lm16;
hence (Ex x,(p => q)) => ((All x,p) => (Ex x,q)) is valid by A2, Th22; :: thesis: ((All x,p) => (Ex x,q)) => (Ex x,(p => q)) is valid
(All x,(p '&' ('not' q))) => ((All x,p) '&' (All x,('not' q))) is valid by Th40;
then A3: ('not' ((All x,p) '&' (All x,('not' q)))) => ('not' (All x,(p '&' ('not' q)))) is valid by LUKASI_1:63;
('not' (All x,(p '&' ('not' q)))) => (Ex x,('not' (p '&' ('not' q)))) is valid by Th55;
then ('not' ((All x,p) '&' (All x,('not' q)))) => (Ex x,('not' (p '&' ('not' q)))) is valid by A3, LUKASI_1:43;
then A4: ('not' ((All x,p) '&' (All x,('not' q)))) => (Ex x,(p => q)) is valid by QC_LANG2:def 2;
(All x,('not' q)) => ('not' ('not' (All x,('not' q)))) is valid by LUKASI_1:64;
then A5: ((All x,p) '&' (All x,('not' q))) => ((All x,p) '&' ('not' ('not' (All x,('not' q))))) is valid by Lm9;
(All x,p) => (Ex x,q) = (All x,p) => ('not' (All x,('not' q))) by QC_LANG2:def 5
.= 'not' ((All x,p) '&' ('not' ('not' (All x,('not' q))))) by QC_LANG2:def 2 ;
then ((All x,p) => (Ex x,q)) => ('not' ((All x,p) '&' (All x,('not' q)))) is valid by A5, LUKASI_1:63;
hence ((All x,p) => (Ex x,q)) => (Ex x,(p => q)) is valid by A4, LUKASI_1:43; :: thesis: verum