:: Propositional Calculus
:: by Grzegorz Bancerek, Agata Darmochwa\l and Andrzej Trybulec
::
:: Received September 26, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

theorem Th1: :: LUKASI_1:1

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A holds (p => q) => ((q => r) => (p => r)) in TAUT A

proof end;

theorem Th2: :: LUKASI_1:2

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => q in TAUT A holds
(q => r) => (p => r) in TAUT A

proof end;

theorem Th3: :: LUKASI_1:3

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => q in TAUT A & q => r in TAUT A holds
p => r in TAUT A

proof end;

theorem Th4: :: LUKASI_1:4

for A being QC-alphabet
for p being Element of CQC-WFF A holds p => p in TAUT A

proof end;

Lm1: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds (((q => r) => (p => r)) => s) => ((p => q) => s) in TAUT A

proof end;

Lm2: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds (p => (q => r)) => ((s => q) => (p => (s => r))) in TAUT A

proof end;

Lm3: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds (p => q) => (((p => r) => s) => ((q => r) => s)) in TAUT A

proof end;

Lm4: for A being QC-alphabet
for p, q, r, s, t being Element of CQC-WFF A holds (t => ((p => r) => s)) => ((p => q) => (t => ((q => r) => s))) in TAUT A

proof end;

Lm5: for A being QC-alphabet
for p, q, r being Element of CQC-WFF A holds ((('not' p) => q) => r) => (p => r) in TAUT A

proof end;

Lm6: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds p => (((('not' p) => r) => s) => ((q => r) => s)) in TAUT A

proof end;

Lm7: for A being QC-alphabet
for p, q being Element of CQC-WFF A holds (q => ((('not' p) => p) => p)) => ((('not' p) => p) => p) in TAUT A

proof end;

Lm8: for A being QC-alphabet
for p, t being Element of CQC-WFF A holds t => ((('not' p) => p) => p) in TAUT A

proof end;

Lm9: for A being QC-alphabet
for p, q, t being Element of CQC-WFF A holds (('not' p) => q) => (t => ((q => p) => p)) in TAUT A

proof end;

Lm10: for A being QC-alphabet
for p, q, r, t being Element of CQC-WFF A holds ((t => ((q => p) => p)) => r) => ((('not' p) => q) => r) in TAUT A

proof end;

Lm11: for A being QC-alphabet
for p, q being Element of CQC-WFF A holds (('not' p) => q) => ((q => p) => p) in TAUT A

proof end;

Lm12: for A being QC-alphabet
for p, q being Element of CQC-WFF A holds p => ((q => p) => p) in TAUT A

proof end;

theorem Th5: :: LUKASI_1:5

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds q => (p => q) in TAUT A

proof end;

theorem Th6: :: LUKASI_1:6

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A holds ((p => q) => r) => (q => r) in TAUT A

proof end;

theorem Th7: :: LUKASI_1:7

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds q => ((q => p) => p) in TAUT A

proof end;

theorem Th8: :: LUKASI_1:8

for A being QC-alphabet
for p, q, s being Element of CQC-WFF A holds (s => (q => p)) => (q => (s => p)) in TAUT A

proof end;

theorem Th9: :: LUKASI_1:9

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A holds (q => r) => ((p => q) => (p => r)) in TAUT A

proof end;

Lm13: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds ((q => (s => p)) => r) => ((s => (q => p)) => r) in TAUT A

proof end;

Lm14: for A being QC-alphabet
for p, q being Element of CQC-WFF A holds ((p => q) => p) => p in TAUT A

proof end;

Lm15: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds ((p => r) => s) => ((p => q) => ((q => r) => s)) in TAUT A

proof end;

Lm16: for A being QC-alphabet
for p, q, r being Element of CQC-WFF A holds ((p => q) => r) => ((r => p) => p) in TAUT A

proof end;

Lm17: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds (((r => p) => p) => s) => (((p => q) => r) => s) in TAUT A

proof end;

Lm18: for A being QC-alphabet
for p, q, r being Element of CQC-WFF A holds ((q => r) => p) => ((q => p) => p) in TAUT A

proof end;

theorem Th10: :: LUKASI_1:10

for A being QC-alphabet
for q, r being Element of CQC-WFF A holds (q => (q => r)) => (q => r) in TAUT A

proof end;

Lm19: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds (q => s) => (((q => r) => p) => ((s => p) => p)) in TAUT A

proof end;

Lm20: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds ((q => r) => p) => ((q => s) => ((s => p) => p)) in TAUT A

proof end;

Lm21: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds (q => s) => ((s => (p => (q => r))) => (p => (q => r))) in TAUT A

proof end;

Lm22: for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A holds (s => (p => (q => r))) => ((q => s) => (p => (q => r))) in TAUT A

proof end;

theorem Th11: :: LUKASI_1:11

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A holds (p => (q => r)) => ((p => q) => (p => r)) in TAUT A

proof end;

theorem Th12: :: LUKASI_1:12

for A being QC-alphabet
for p being Element of CQC-WFF A holds ('not' (VERUM A)) => p in TAUT A

proof end;

theorem Th13: :: LUKASI_1:13

for A being QC-alphabet
for p, q being Element of CQC-WFF A st q in TAUT A holds
p => q in TAUT A

proof end;

theorem :: LUKASI_1:14

for A being QC-alphabet
for p, q being Element of CQC-WFF A st p in TAUT A holds
(p => q) => q in TAUT A

proof end;

theorem Th15: :: LUKASI_1:15

for A being QC-alphabet
for p, q, s being Element of CQC-WFF A st s => (q => p) in TAUT A holds
q => (s => p) in TAUT A

proof end;

theorem Th16: :: LUKASI_1:16

for A being QC-alphabet
for p, q, s being Element of CQC-WFF A st s => (q => p) in TAUT A & q in TAUT A holds
s => p in TAUT A

proof end;

theorem :: LUKASI_1:17

for A being QC-alphabet
for p, q, s being Element of CQC-WFF A st s => (q => p) in TAUT A & q in TAUT A & s in TAUT A holds
p in TAUT A

proof end;

theorem :: LUKASI_1:18

for A being QC-alphabet
for q, r being Element of CQC-WFF A st q => (q => r) in TAUT A holds
q => r in TAUT A

proof end;

theorem Th19: :: LUKASI_1:19

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A holds
(p => q) => (p => r) in TAUT A

proof end;

theorem Th20: :: LUKASI_1:20

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A & p => q in TAUT A holds
p => r in TAUT A

proof end;

theorem :: LUKASI_1:21

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A & p => q in TAUT A & p in TAUT A holds
r in TAUT A

proof end;

theorem Th22: :: LUKASI_1:22

for A being QC-alphabet
for p, q, r, s being Element of CQC-WFF A st p => (q => r) in TAUT A & p => (r => s) in TAUT A holds
p => (q => s) in TAUT A

proof end;

theorem :: LUKASI_1:23

for A being QC-alphabet
for p being Element of CQC-WFF A holds p => (VERUM A) in TAUT A by Th13, CQC_THE1:41;

Lm23: for A being QC-alphabet
for p being Element of CQC-WFF A holds ('not' p) => (p => ('not' (VERUM A))) in TAUT A

proof end;

Lm24: for A being QC-alphabet
for p being Element of CQC-WFF A holds (('not' p) => ('not' (VERUM A))) => p in TAUT A

proof end;

theorem Th24: :: LUKASI_1:24

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds (('not' p) => ('not' q)) => (q => p) in TAUT A

proof end;

theorem Th25: :: LUKASI_1:25

for A being QC-alphabet
for p being Element of CQC-WFF A holds ('not' ('not' p)) => p in TAUT A

proof end;

Lm25: now :: thesis: for A being QC-alphabet
for p being Element of CQC-WFF A holds (p => ('not' (VERUM A))) => ('not' p) in TAUT A

let A be QC-alphabet ; :: thesis: for p being Element of CQC-WFF A holds (p => ('not' (VERUM A))) => ('not' p) in TAUT A
let p be Element of CQC-WFF A; :: thesis: (p => ('not' (VERUM A))) => ('not' p) in TAUT A
('not' ('not' p)) => p in TAUT A by Th25;
then A1: (p => ('not' (VERUM A))) => (('not' ('not' p)) => ('not' (VERUM A))) in TAUT A by Th2;
(('not' ('not' p)) => ('not' (VERUM A))) => ('not' p) in TAUT A by Lm24;
hence (p => ('not' (VERUM A))) => ('not' p) in TAUT A by A1, Th3; :: thesis: verum

end;

theorem Th26: :: LUKASI_1:26

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds (p => q) => (('not' q) => ('not' p)) in TAUT A

proof end;

theorem Th27: :: LUKASI_1:27

for A being QC-alphabet
for p being Element of CQC-WFF A holds p => ('not' ('not' p)) in TAUT A

proof end;

theorem Th28: :: LUKASI_1:28

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( (('not' ('not' p)) => q) => (p => q) in TAUT A & (p => q) => (('not' ('not' p)) => q) in TAUT A )

proof end;

theorem Th29: :: LUKASI_1:29

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( (p => ('not' ('not' q))) => (p => q) in TAUT A & (p => q) => (p => ('not' ('not' q))) in TAUT A )

proof end;

theorem Th30: :: LUKASI_1:30

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds (p => ('not' q)) => (q => ('not' p)) in TAUT A

proof end;

theorem Th31: :: LUKASI_1:31

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds (('not' p) => q) => (('not' q) => p) in TAUT A

proof end;

theorem :: LUKASI_1:32

for A being QC-alphabet
for p being Element of CQC-WFF A holds (p => ('not' p)) => ('not' p) in TAUT A

proof end;

theorem :: LUKASI_1:33

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds ('not' p) => (p => q) in TAUT A

proof end;

theorem Th34: :: LUKASI_1:34

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( p => q in TAUT A iff ('not' q) => ('not' p) in TAUT A )

proof end;

theorem :: LUKASI_1:35

for A being QC-alphabet
for p, q being Element of CQC-WFF A st ('not' p) => ('not' q) in TAUT A holds
q => p in TAUT A by Th34;

theorem :: LUKASI_1:36

for A being QC-alphabet
for p being Element of CQC-WFF A holds
( p in TAUT A iff 'not' ('not' p) in TAUT A )

proof end;

theorem :: LUKASI_1:37

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( p => q in TAUT A iff p => ('not' ('not' q)) in TAUT A )

proof end;

theorem :: LUKASI_1:38

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( p => q in TAUT A iff ('not' ('not' p)) => q in TAUT A )

proof end;

theorem :: LUKASI_1:39

for A being QC-alphabet
for p, q being Element of CQC-WFF A st p => ('not' q) in TAUT A holds
q => ('not' p) in TAUT A

proof end;

theorem :: LUKASI_1:40

for A being QC-alphabet
for p, q being Element of CQC-WFF A st ('not' p) => q in TAUT A holds
('not' q) => p in TAUT A

proof end;

:: predykat |- i schematy konsekwencji

registration

let A be QC-alphabet ;
let p, q, r be Element of CQC-WFF A;

cluster K167(A,(p => q),((q => r) => (p => r))) -> valid ;

coherence
(p => q) => ((q => r) => (p => r)) is valid by Th1;

end;

theorem :: LUKASI_1:41

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => q is valid holds
(q => r) => (p => r) is valid

proof end;

theorem Th42: :: LUKASI_1:42

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => q is valid & q => r is valid holds
p => r is valid by Th3;

registration

let A be QC-alphabet ;
let p be Element of CQC-WFF A;

cluster K167(A,p,p) -> valid ;

coherence
p => p is valid by Th4;

end;

registration

let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;

cluster K167(A,p,(q => p)) -> valid ;

coherence
p => (q => p) is valid by Th5;

end;

theorem :: LUKASI_1:43

for A being QC-alphabet
for p, q being Element of CQC-WFF A st p is valid holds
q => p is valid by Th13;

registration

let A be QC-alphabet ;
let p, q, s be Element of CQC-WFF A;

cluster K167(A,(s => (q => p)),(q => (s => p))) -> valid ;

coherence
(s => (q => p)) => (q => (s => p)) is valid by Th8;

end;

theorem Th44: :: LUKASI_1:44

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => (q => r) is valid holds
q => (p => r) is valid by Th15;

theorem :: LUKASI_1:45

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => (q => r) is valid & q is valid holds
p => r is valid

proof end;

theorem :: LUKASI_1:46

for A being QC-alphabet
for p being Element of CQC-WFF A holds
( p => (VERUM A) is valid & ('not' (VERUM A)) => p is valid ) by Th13, CQC_THE1:41, Th12;

registration

let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;

cluster K167(A,p,((p => q) => q)) -> valid ;

coherence
p => ((p => q) => q) is valid by Th7;

end;

registration

let A be QC-alphabet ;
let q, r be Element of CQC-WFF A;

cluster K167(A,(q => (q => r)),(q => r)) -> valid ;

coherence
(q => (q => r)) => (q => r) is valid by Th10;

end;

theorem :: LUKASI_1:47

for A being QC-alphabet
for q, r being Element of CQC-WFF A st q => (q => r) is valid holds
q => r is valid

proof end;

registration

let A be QC-alphabet ;
let p, q, r be Element of CQC-WFF A;

cluster K167(A,(p => (q => r)),((p => q) => (p => r))) -> valid ;

coherence
(p => (q => r)) => ((p => q) => (p => r)) is valid by Th11;

end;

theorem Th48: :: LUKASI_1:48

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => (q => r) is valid holds
(p => q) => (p => r) is valid

proof end;

theorem :: LUKASI_1:49

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => (q => r) is valid & p => q is valid holds
p => r is valid

proof end;

registration

let A be QC-alphabet ;
let p, q, r be Element of CQC-WFF A;

cluster K167(A,((p => q) => r),(q => r)) -> valid ;

coherence
((p => q) => r) => (q => r) is valid by Th6;

end;

theorem :: LUKASI_1:50

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st (p => q) => r is valid holds
q => r is valid

proof end;

registration

let A be QC-alphabet ;
let p, q, r be Element of CQC-WFF A;

cluster K167(A,(p => q),((r => p) => (r => q))) -> valid ;

coherence
(p => q) => ((r => p) => (r => q)) is valid by Th9;

end;

theorem :: LUKASI_1:51

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p => q is valid holds
(r => p) => (r => q) is valid

proof end;

registration

let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;

cluster K167(A,(p => q),(('not' q) => ('not' p))) -> valid ;

coherence
(p => q) => (('not' q) => ('not' p)) is valid by Th26;

end;

registration

let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;

cluster K167(A,(('not' p) => ('not' q)),(q => p)) -> valid ;

coherence
(('not' p) => ('not' q)) => (q => p) is valid by Th24;

end;

theorem :: LUKASI_1:52

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( ('not' p) => ('not' q) is valid iff q => p is valid )

proof end;

registration

let A be QC-alphabet ;
let p be Element of CQC-WFF A;

cluster K167(A,p,('not' ('not' p))) -> valid ;

coherence
p => ('not' ('not' p)) is valid by Th27;

end;

registration

let A be QC-alphabet ;
let p be Element of CQC-WFF A;

cluster K167(A,('not' ('not' p)),p) -> valid ;

coherence
('not' ('not' p)) => p is valid by Th25;

end;

theorem :: LUKASI_1:53

for A being QC-alphabet
for p being Element of CQC-WFF A holds
( 'not' ('not' p) is valid iff p is valid )

proof end;

registration

let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;

cluster K167(A,(('not' ('not' p)) => q),(p => q)) -> valid ;

coherence
(('not' ('not' p)) => q) => (p => q) is valid by Th28;

end;

theorem :: LUKASI_1:54

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( ('not' ('not' p)) => q is valid iff p => q is valid )

proof end;

registration

let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;

cluster K167(A,(p => ('not' ('not' q))),(p => q)) -> valid ;

coherence
(p => ('not' ('not' q))) => (p => q) is valid by Th29;

end;

theorem :: LUKASI_1:55

for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( p => ('not' ('not' q)) is valid iff p => q is valid )

proof end;

registration

let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;

cluster K167(A,(p => ('not' q)),(q => ('not' p))) -> valid ;

coherence
(p => ('not' q)) => (q => ('not' p)) is valid by Th30;

end;

theorem :: LUKASI_1:56

for A being QC-alphabet
for p, q being Element of CQC-WFF A st p => ('not' q) is valid holds
q => ('not' p) is valid

proof end;

registration

let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;

cluster K167(A,(('not' p) => q),(('not' q) => p)) -> valid ;

coherence
(('not' p) => q) => (('not' q) => p) is valid by Th31;

end;

theorem :: LUKASI_1:57

for A being QC-alphabet
for p, q being Element of CQC-WFF A st ('not' p) => q is valid holds
('not' q) => p is valid

proof end;

theorem :: LUKASI_1:58

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => q holds
X |- (q => r) => (p => r)

proof end;

theorem Th59: :: LUKASI_1:59

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => q & X |- q => r holds
X |- p => r

proof end;

theorem :: LUKASI_1:60

for A being QC-alphabet
for p being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) holds X |- p => p by CQC_THE1:59;

theorem :: LUKASI_1:61

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p holds
X |- q => p

proof end;

theorem :: LUKASI_1:62

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p holds
X |- (p => q) => q

proof end;

theorem Th63: :: LUKASI_1:63

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => (q => r) holds
X |- q => (p => r)

proof end;

theorem :: LUKASI_1:64

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => (q => r) & X |- q holds
X |- p => r

proof end;

theorem :: LUKASI_1:65

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => (p => q) holds
X |- p => q

proof end;

theorem :: LUKASI_1:66

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- (p => q) => r holds
X |- q => r

proof end;

theorem Th67: :: LUKASI_1:67

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => (q => r) holds
X |- (p => q) => (p => r)

proof end;

theorem :: LUKASI_1:68

for A being QC-alphabet
for p, q, r being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => (q => r) & X |- p => q holds
X |- p => r

proof end;

theorem :: LUKASI_1:69

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) holds
( X |- ('not' p) => ('not' q) iff X |- q => p )

proof end;

theorem :: LUKASI_1:70

for A being QC-alphabet
for p being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) holds
( X |- 'not' ('not' p) iff X |- p )

proof end;

theorem :: LUKASI_1:71

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) holds
( X |- p => ('not' ('not' q)) iff X |- p => q )

proof end;

theorem :: LUKASI_1:72

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) holds
( X |- ('not' ('not' p)) => q iff X |- p => q )

proof end;

theorem Th73: :: LUKASI_1:73

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => ('not' q) holds
X |- q => ('not' p)

proof end;

theorem Th74: :: LUKASI_1:74

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- ('not' p) => q holds
X |- ('not' q) => p

proof end;

theorem :: LUKASI_1:75

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- p => ('not' q) & X |- q holds
X |- 'not' p

proof end;

theorem :: LUKASI_1:76

for A being QC-alphabet
for p, q being Element of CQC-WFF A
for X being Subset of (CQC-WFF A) st X |- ('not' p) => q & X |- 'not' q holds
X |- p

proof end;