:: by Agata Darmochwa\l

::

:: Received October 24, 1990

:: Copyright (c) 1990-2021 Association of Mizar Users

theorem Th1: :: CQC_THE2:1

for A being QC-alphabet

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

(p '&' q) => r is valid

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

(p '&' q) => r is valid

proof end;

theorem Th2: :: CQC_THE2:2

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

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

(q '&' p) => r is valid

proof end;

theorem Th3: :: CQC_THE2:3

for A being QC-alphabet

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

p => (q => r) is valid

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

p => (q => r) is valid

proof end;

theorem Th4: :: CQC_THE2:4

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

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

q => (p => r) is valid

proof end;

Lm1: for A being QC-alphabet

for p, q being Element of CQC-WFF A holds

( (p '&' q) => p is valid & (p '&' q) => q is valid )

by PROCAL_1:19, PROCAL_1:21;

Lm2: for A being QC-alphabet

for p, q being Element of CQC-WFF A st p '&' q is valid holds

( p is valid & q is valid )

proof end;

Lm3: for A being QC-alphabet

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

p => (q '&' r) is valid

by PROCAL_1:52;

Lm4: for A being QC-alphabet

for p, q, r, t being Element of CQC-WFF A st p => q is valid & r => t is valid holds

(p 'or' r) => (q 'or' t) is valid

by PROCAL_1:57;

Lm5: for A being QC-alphabet

for p, q, r, t being Element of CQC-WFF A st p => q is valid & r => t is valid holds

(p '&' r) => (q '&' t) is valid

by PROCAL_1:56;

Lm6: for A being QC-alphabet

for p, q being Element of CQC-WFF A holds

( p => (p 'or' q) is valid & p => (q 'or' p) is valid )

by PROCAL_1:3, PROCAL_1:4;

Lm7: for A being QC-alphabet

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

(p 'or' r) => q is valid

by PROCAL_1:53;

Lm8: for A being QC-alphabet

for p, q being Element of CQC-WFF A st p is valid & q is valid holds

p '&' q is valid

by PROCAL_1:47;

Lm9: 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

by PROCAL_1:50;

Lm10: for A being QC-alphabet

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

(p 'or' r) => (q 'or' r) is valid

by PROCAL_1:48;

Lm11: for A being QC-alphabet

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

by PROCAL_1:5;

Lm12: for A being QC-alphabet

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

proof end;

Lm13: for A being QC-alphabet

for p being Element of CQC-WFF A holds p <=> p is valid

proof end;

Lm14: for A being QC-alphabet

for p, q being Element of CQC-WFF A holds

( ( p => q is valid & q => p is valid ) iff p <=> q is valid )

proof end;

Lm15: for A being QC-alphabet

for p, q being Element of CQC-WFF A st p <=> q is valid holds

( p is valid iff q is valid )

proof end;

Lm16: for A being QC-alphabet

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

p => (q => t) is valid

proof end;

::------------------------------------------------

theorem Th5: :: CQC_THE2:5

for A being QC-alphabet

for s being QC-formula of A

for x, y being bound_QC-variable of A holds

( y in still_not-bound_in (All (x,s)) iff ( y in still_not-bound_in s & y <> x ) )

for s being QC-formula of A

for x, y being bound_QC-variable of A holds

( y in still_not-bound_in (All (x,s)) iff ( y in still_not-bound_in s & y <> x ) )

proof end;

theorem Th6: :: CQC_THE2:6

for A being QC-alphabet

for s being QC-formula of A

for x, y being bound_QC-variable of A holds

( y in still_not-bound_in (Ex (x,s)) iff ( y in still_not-bound_in s & y <> x ) )

for s being QC-formula of A

for x, y being bound_QC-variable of A holds

( y in still_not-bound_in (Ex (x,s)) iff ( y in still_not-bound_in s & y <> x ) )

proof end;

theorem Th7: :: CQC_THE2:7

for A being QC-alphabet

for s, h being QC-formula of A

for y being bound_QC-variable of A holds

( y in still_not-bound_in (s => h) iff ( y in still_not-bound_in s or y in still_not-bound_in h ) )

for s, h being QC-formula of A

for y being bound_QC-variable of A holds

( y in still_not-bound_in (s => h) iff ( y in still_not-bound_in s or y in still_not-bound_in h ) )

proof end;

theorem Th8: :: CQC_THE2:8

for A being QC-alphabet

for s, h being QC-formula of A

for y being bound_QC-variable of A holds

( y in still_not-bound_in (s '&' h) iff ( y in still_not-bound_in s or y in still_not-bound_in h ) )

for s, h being QC-formula of A

for y being bound_QC-variable of A holds

( y in still_not-bound_in (s '&' h) iff ( y in still_not-bound_in s or y in still_not-bound_in h ) )

proof end;

theorem Th9: :: CQC_THE2:9

for A being QC-alphabet

for s, h being QC-formula of A

for y being bound_QC-variable of A holds

( y in still_not-bound_in (s 'or' h) iff ( y in still_not-bound_in s or y in still_not-bound_in h ) )

for s, h being QC-formula of A

for y being bound_QC-variable of A holds

( y in still_not-bound_in (s 'or' h) iff ( y in still_not-bound_in s or y in still_not-bound_in h ) )

proof end;

theorem :: CQC_THE2:10

for A being QC-alphabet

for s being QC-formula of A

for x, y being bound_QC-variable of A holds

( not x in still_not-bound_in (All (x,y,s)) & not y in still_not-bound_in (All (x,y,s)) )

for s being QC-formula of A

for x, y being bound_QC-variable of A holds

( not x in still_not-bound_in (All (x,y,s)) & not y in still_not-bound_in (All (x,y,s)) )

proof end;

theorem :: CQC_THE2:11

for A being QC-alphabet

for s being QC-formula of A

for x, y being bound_QC-variable of A holds

( not x in still_not-bound_in (Ex (x,y,s)) & not y in still_not-bound_in (Ex (x,y,s)) )

for s being QC-formula of A

for x, y being bound_QC-variable of A holds

( not x in still_not-bound_in (Ex (x,y,s)) & not y in still_not-bound_in (Ex (x,y,s)) )

proof end;

theorem Th12: :: CQC_THE2:12

for A being QC-alphabet

for s, h being QC-formula of A

for x being bound_QC-variable of A holds (s => h) . x = (s . x) => (h . x)

for s, h being QC-formula of A

for x being bound_QC-variable of A holds (s => h) . x = (s . x) => (h . x)

proof end;

theorem :: CQC_THE2:13

for A being QC-alphabet

for s, h being QC-formula of A

for x being bound_QC-variable of A holds (s 'or' h) . x = (s . x) 'or' (h . x)

for s, h being QC-formula of A

for x being bound_QC-variable of A holds (s 'or' h) . x = (s . x) 'or' (h . x)

proof end;

theorem :: CQC_THE2:14

for A being QC-alphabet

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A st x <> y holds

(Ex (x,p)) . y = Ex (x,(p . y))

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A st x <> y holds

(Ex (x,p)) . y = Ex (x,(p . y))

proof end;

::---------------------------------------------------------

theorem Th15: :: CQC_THE2:15

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds p => (Ex (x,p)) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds p => (Ex (x,p)) is valid

proof end;

theorem Th16: :: CQC_THE2:16

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st p is valid holds

Ex (x,p) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st p is valid holds

Ex (x,p) is valid

proof end;

theorem :: CQC_THE2:17

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,p)) => (Ex (x,p)) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,p)) => (Ex (x,p)) is valid

proof end;

theorem :: CQC_THE2:18

for A being QC-alphabet

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A holds (All (x,p)) => (Ex (y,p)) is valid

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A holds (All (x,p)) => (Ex (y,p)) is valid

proof end;

theorem Th19: :: CQC_THE2:19

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st p => q is valid & not x in still_not-bound_in q holds

(Ex (x,p)) => q is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st p => q is valid & not x in still_not-bound_in q holds

(Ex (x,p)) => q is valid

proof end;

theorem Th20: :: CQC_THE2:20

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(Ex (x,p)) => p is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(Ex (x,p)) => p is valid

proof end;

theorem :: CQC_THE2:21

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p & Ex (x,p) is valid holds

p is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p & Ex (x,p) is valid holds

p is valid

proof end;

theorem Th22: :: CQC_THE2:22

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not y in still_not-bound_in h holds

p => (Ex (y,q)) is valid

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not y in still_not-bound_in h holds

p => (Ex (y,q)) is valid

proof end;

theorem Th23: :: CQC_THE2:23

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st p is valid holds

All (x,p) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st p is valid holds

All (x,p) is valid

proof end;

theorem Th24: :: CQC_THE2:24

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

p => (All (x,p)) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

p => (All (x,p)) is valid

proof end;

theorem Th25: :: CQC_THE2:25

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not x in still_not-bound_in h holds

(All (x,p)) => q is valid

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not x in still_not-bound_in h holds

(All (x,p)) => q is valid

proof end;

theorem :: CQC_THE2:26

for A being QC-alphabet

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A st not y in still_not-bound_in p holds

(All (x,p)) => (All (y,p)) is valid

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A st not y in still_not-bound_in p holds

(All (x,p)) => (All (y,p)) is valid

proof end;

theorem :: CQC_THE2:27

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not x in still_not-bound_in h & not y in still_not-bound_in p holds

(All (x,p)) => (All (y,q)) is valid

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not x in still_not-bound_in h & not y in still_not-bound_in p holds

(All (x,p)) => (All (y,q)) is valid

proof end;

theorem :: CQC_THE2:28

for A being QC-alphabet

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A st not x in still_not-bound_in p holds

(Ex (x,p)) => (Ex (y,p)) is valid

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A st not x in still_not-bound_in p holds

(Ex (x,p)) => (Ex (y,p)) is valid

proof end;

theorem :: CQC_THE2:29

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not x in still_not-bound_in q & not y in still_not-bound_in h holds

(Ex (x,p)) => (Ex (y,q)) is valid

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not x in still_not-bound_in q & not y in still_not-bound_in h holds

(Ex (x,p)) => (Ex (y,q)) is valid

proof end;

theorem Th30: :: CQC_THE2:30

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p => q))) => ((All (x,p)) => (All (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p => q))) => ((All (x,p)) => (All (x,q))) is valid

proof end;

theorem Th31: :: CQC_THE2:31

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st All (x,(p => q)) is valid holds

(All (x,p)) => (All (x,q)) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st All (x,(p => q)) is valid holds

(All (x,p)) => (All (x,q)) is valid

proof end;

theorem Th32: :: CQC_THE2:32

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p <=> q))) => ((All (x,p)) <=> (All (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p <=> q))) => ((All (x,p)) <=> (All (x,q))) is valid

proof end;

theorem :: CQC_THE2:33

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st All (x,(p <=> q)) is valid holds

(All (x,p)) <=> (All (x,q)) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st All (x,(p <=> q)) is valid holds

(All (x,p)) <=> (All (x,q)) is valid

proof end;

theorem Th34: :: CQC_THE2:34

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p => q))) => ((Ex (x,p)) => (Ex (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p => q))) => ((Ex (x,p)) => (Ex (x,q))) is valid

proof end;

theorem Th35: :: CQC_THE2:35

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st All (x,(p => q)) is valid holds

(Ex (x,p)) => (Ex (x,q)) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st All (x,(p => q)) is valid holds

(Ex (x,p)) => (Ex (x,q)) is valid

proof end;

theorem Th36: :: CQC_THE2:36

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( (All (x,(p '&' q))) => ((All (x,p)) '&' (All (x,q))) is valid & ((All (x,p)) '&' (All (x,q))) => (All (x,(p '&' q))) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( (All (x,(p '&' q))) => ((All (x,p)) '&' (All (x,q))) is valid & ((All (x,p)) '&' (All (x,q))) => (All (x,(p '&' q))) is valid )

proof end;

theorem Th37: :: CQC_THE2:37

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p '&' q))) <=> ((All (x,p)) '&' (All (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p '&' q))) <=> ((All (x,p)) '&' (All (x,q))) is valid

proof end;

theorem :: CQC_THE2:38

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( All (x,(p '&' q)) is valid iff (All (x,p)) '&' (All (x,q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( All (x,(p '&' q)) is valid iff (All (x,p)) '&' (All (x,q)) is valid )

proof end;

theorem Th39: :: CQC_THE2:39

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds ((All (x,p)) 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds ((All (x,p)) 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid

proof end;

theorem Th40: :: CQC_THE2:40

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( (Ex (x,(p 'or' q))) => ((Ex (x,p)) 'or' (Ex (x,q))) is valid & ((Ex (x,p)) 'or' (Ex (x,q))) => (Ex (x,(p 'or' q))) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( (Ex (x,(p 'or' q))) => ((Ex (x,p)) 'or' (Ex (x,q))) is valid & ((Ex (x,p)) 'or' (Ex (x,q))) => (Ex (x,(p 'or' q))) is valid )

proof end;

theorem Th41: :: CQC_THE2:41

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (Ex (x,(p 'or' q))) <=> ((Ex (x,p)) 'or' (Ex (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (Ex (x,(p 'or' q))) <=> ((Ex (x,p)) 'or' (Ex (x,q))) is valid

proof end;

theorem :: CQC_THE2:42

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( Ex (x,(p 'or' q)) is valid iff (Ex (x,p)) 'or' (Ex (x,q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( Ex (x,(p 'or' q)) is valid iff (Ex (x,p)) 'or' (Ex (x,q)) is valid )

proof end;

theorem Th43: :: CQC_THE2:43

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (Ex (x,(p '&' q))) => ((Ex (x,p)) '&' (Ex (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (Ex (x,(p '&' q))) => ((Ex (x,p)) '&' (Ex (x,q))) is valid

proof end;

theorem :: CQC_THE2:44

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st Ex (x,(p '&' q)) is valid holds

(Ex (x,p)) '&' (Ex (x,q)) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st Ex (x,(p '&' q)) is valid holds

(Ex (x,p)) '&' (Ex (x,q)) is valid

proof end;

theorem Th45: :: CQC_THE2:45

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( (All (x,('not' ('not' p)))) => (All (x,p)) is valid & (All (x,p)) => (All (x,('not' ('not' p)))) is valid )

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( (All (x,('not' ('not' p)))) => (All (x,p)) is valid & (All (x,p)) => (All (x,('not' ('not' p)))) is valid )

proof end;

theorem :: CQC_THE2:46

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,('not' ('not' p)))) <=> (All (x,p)) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,('not' ('not' p)))) <=> (All (x,p)) is valid

proof end;

theorem Th47: :: CQC_THE2:47

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( (Ex (x,('not' ('not' p)))) => (Ex (x,p)) is valid & (Ex (x,p)) => (Ex (x,('not' ('not' p)))) is valid )

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( (Ex (x,('not' ('not' p)))) => (Ex (x,p)) is valid & (Ex (x,p)) => (Ex (x,('not' ('not' p)))) is valid )

proof end;

theorem :: CQC_THE2:48

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds (Ex (x,('not' ('not' p)))) <=> (Ex (x,p)) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds (Ex (x,('not' ('not' p)))) <=> (Ex (x,p)) is valid

proof end;

theorem Th49: :: CQC_THE2:49

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( ('not' (Ex (x,('not' p)))) => (All (x,p)) is valid & (All (x,p)) => ('not' (Ex (x,('not' p)))) is valid )

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( ('not' (Ex (x,('not' p)))) => (All (x,p)) is valid & (All (x,p)) => ('not' (Ex (x,('not' p)))) is valid )

proof end;

theorem :: CQC_THE2:50

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds ('not' (Ex (x,('not' p)))) <=> (All (x,p)) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds ('not' (Ex (x,('not' p)))) <=> (All (x,p)) is valid

proof end;

theorem Th51: :: CQC_THE2:51

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( ('not' (All (x,p))) => (Ex (x,('not' p))) is valid & (Ex (x,('not' p))) => ('not' (All (x,p))) is valid )

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( ('not' (All (x,p))) => (Ex (x,('not' p))) is valid & (Ex (x,('not' p))) => ('not' (All (x,p))) is valid )

proof end;

theorem :: CQC_THE2:52

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds ('not' (All (x,p))) <=> (Ex (x,('not' p))) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds ('not' (All (x,p))) <=> (Ex (x,('not' p))) is valid

proof end;

theorem :: CQC_THE2:53

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( ('not' (Ex (x,p))) => (All (x,('not' p))) is valid & (All (x,('not' p))) => ('not' (Ex (x,p))) is valid )

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( ('not' (Ex (x,p))) => (All (x,('not' p))) is valid & (All (x,('not' p))) => ('not' (Ex (x,p))) is valid )

proof end;

theorem Th54: :: CQC_THE2:54

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,('not' p))) <=> ('not' (Ex (x,p))) is valid

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,('not' p))) <=> ('not' (Ex (x,p))) is valid

proof end;

theorem :: CQC_THE2:55

for A being QC-alphabet

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A holds

( (All (x,(All (y,p)))) => (All (y,(All (x,p)))) is valid & (All (x,y,p)) => (All (y,x,p)) is valid )

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A holds

( (All (x,(All (y,p)))) => (All (y,(All (x,p)))) is valid & (All (x,y,p)) => (All (y,x,p)) is valid )

proof end;

theorem :: CQC_THE2:56

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not y in still_not-bound_in h holds

(All (x,(All (y,q)))) => (All (x,p)) is valid

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not y in still_not-bound_in h holds

(All (x,(All (y,q)))) => (All (x,p)) is valid

proof end;

theorem :: CQC_THE2:57

for A being QC-alphabet

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A holds

( (Ex (x,(Ex (y,p)))) => (Ex (y,(Ex (x,p)))) is valid & (Ex (x,y,p)) => (Ex (y,x,p)) is valid )

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A holds

( (Ex (x,(Ex (y,p)))) => (Ex (y,(Ex (x,p)))) is valid & (Ex (x,y,p)) => (Ex (y,x,p)) is valid )

proof end;

theorem :: CQC_THE2:58

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not y in still_not-bound_in h holds

(Ex (x,p)) => (Ex (x,y,q)) is valid

for p, q being Element of CQC-WFF A

for h being QC-formula of A

for x, y being bound_QC-variable of A st p = h . x & q = h . y & not y in still_not-bound_in h holds

(Ex (x,p)) => (Ex (x,y,q)) is valid

proof end;

theorem :: CQC_THE2:59

for A being QC-alphabet

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A holds (Ex (x,(All (y,p)))) => (All (y,(Ex (x,p)))) is valid

for p being Element of CQC-WFF A

for x, y being bound_QC-variable of A holds (Ex (x,(All (y,p)))) => (All (y,(Ex (x,p)))) is valid

proof end;

theorem :: CQC_THE2:60

for A being QC-alphabet

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds Ex (x,(p <=> p)) is valid by Lm13, Th16;

for p being Element of CQC-WFF A

for x being bound_QC-variable of A holds Ex (x,(p <=> p)) is valid by Lm13, Th16;

theorem Th61: :: CQC_THE2:61

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A 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 )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A 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 )

proof end;

theorem Th62: :: CQC_THE2:62

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (Ex (x,(p => q))) <=> ((All (x,p)) => (Ex (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (Ex (x,(p => q))) <=> ((All (x,p)) => (Ex (x,q))) is valid

proof end;

theorem :: CQC_THE2:63

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( Ex (x,(p => q)) is valid iff (All (x,p)) => (Ex (x,q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds

( Ex (x,(p => q)) is valid iff (All (x,p)) => (Ex (x,q)) is valid )

proof end;

theorem Th64: :: CQC_THE2:64

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p '&' q))) => (p '&' (All (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p '&' q))) => (p '&' (All (x,q))) is valid

proof end;

theorem Th65: :: CQC_THE2:65

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p '&' q))) => ((All (x,p)) '&' q) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (All (x,(p '&' q))) => ((All (x,p)) '&' q) is valid

proof end;

theorem Th66: :: CQC_THE2:66

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p '&' (All (x,q))) => (All (x,(p '&' q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p '&' (All (x,q))) => (All (x,(p '&' q))) is valid

proof end;

theorem :: CQC_THE2:67

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p & p '&' (All (x,q)) is valid holds

All (x,(p '&' q)) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p & p '&' (All (x,q)) is valid holds

All (x,(p '&' q)) is valid

proof end;

theorem Th68: :: CQC_THE2:68

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( (p 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid & (All (x,(p 'or' q))) => (p 'or' (All (x,q))) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( (p 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid & (All (x,(p 'or' q))) => (p 'or' (All (x,q))) is valid )

proof end;

theorem Th69: :: CQC_THE2:69

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p 'or' (All (x,q))) <=> (All (x,(p 'or' q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p 'or' (All (x,q))) <=> (All (x,(p 'or' q))) is valid

proof end;

theorem :: CQC_THE2:70

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( p 'or' (All (x,q)) is valid iff All (x,(p 'or' q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( p 'or' (All (x,q)) is valid iff All (x,(p 'or' q)) is valid )

proof end;

theorem Th71: :: CQC_THE2:71

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( (p '&' (Ex (x,q))) => (Ex (x,(p '&' q))) is valid & (Ex (x,(p '&' q))) => (p '&' (Ex (x,q))) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( (p '&' (Ex (x,q))) => (Ex (x,(p '&' q))) is valid & (Ex (x,(p '&' q))) => (p '&' (Ex (x,q))) is valid )

proof end;

theorem Th72: :: CQC_THE2:72

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p '&' (Ex (x,q))) <=> (Ex (x,(p '&' q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p '&' (Ex (x,q))) <=> (Ex (x,(p '&' q))) is valid

proof end;

theorem :: CQC_THE2:73

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( p '&' (Ex (x,q)) is valid iff Ex (x,(p '&' q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( p '&' (Ex (x,q)) is valid iff Ex (x,(p '&' q)) is valid )

proof end;

Lm17: for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(All (x,(p => q))) => (p => (All (x,q))) is valid

proof end;

theorem Th74: :: CQC_THE2:74

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( (All (x,(p => q))) => (p => (All (x,q))) is valid & (p => (All (x,q))) => (All (x,(p => q))) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( (All (x,(p => q))) => (p => (All (x,q))) is valid & (p => (All (x,q))) => (All (x,(p => q))) is valid )

proof end;

theorem Th75: :: CQC_THE2:75

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p => (All (x,q))) <=> (All (x,(p => q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p => (All (x,q))) <=> (All (x,(p => q))) is valid

proof end;

theorem :: CQC_THE2:76

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( All (x,(p => q)) is valid iff p => (All (x,q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( All (x,(p => q)) is valid iff p => (All (x,q)) is valid )

proof end;

theorem Th77: :: CQC_THE2:77

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

(Ex (x,(p => q))) => ((All (x,p)) => q) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

(Ex (x,(p => q))) => ((All (x,p)) => q) is valid

proof end;

theorem Th78: :: CQC_THE2:78

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds ((All (x,p)) => q) => (Ex (x,(p => q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds ((All (x,p)) => q) => (Ex (x,(p => q))) is valid

proof end;

theorem :: CQC_THE2:79

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

( (All (x,p)) => q is valid iff Ex (x,(p => q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

( (All (x,p)) => q is valid iff Ex (x,(p => q)) is valid )

proof end;

theorem Th80: :: CQC_THE2:80

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

( ((Ex (x,p)) => q) => (All (x,(p => q))) is valid & (All (x,(p => q))) => ((Ex (x,p)) => q) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

( ((Ex (x,p)) => q) => (All (x,(p => q))) is valid & (All (x,(p => q))) => ((Ex (x,p)) => q) is valid )

proof end;

theorem Th81: :: CQC_THE2:81

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

((Ex (x,p)) => q) <=> (All (x,(p => q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

((Ex (x,p)) => q) <=> (All (x,(p => q))) is valid

proof end;

theorem :: CQC_THE2:82

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

( (Ex (x,p)) => q is valid iff All (x,(p => q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in q holds

( (Ex (x,p)) => q is valid iff All (x,(p => q)) is valid )

proof end;

theorem Th83: :: CQC_THE2:83

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(Ex (x,(p => q))) => (p => (Ex (x,q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(Ex (x,(p => q))) => (p => (Ex (x,q))) is valid

proof end;

theorem Th84: :: CQC_THE2:84

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (p => (Ex (x,q))) => (Ex (x,(p => q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A holds (p => (Ex (x,q))) => (Ex (x,(p => q))) is valid

proof end;

theorem Th85: :: CQC_THE2:85

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p => (Ex (x,q))) <=> (Ex (x,(p => q))) is valid

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

(p => (Ex (x,q))) <=> (Ex (x,(p => q))) is valid

proof end;

theorem :: CQC_THE2:86

for A being QC-alphabet

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( p => (Ex (x,q)) is valid iff Ex (x,(p => q)) is valid )

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

( p => (Ex (x,q)) is valid iff Ex (x,(p => q)) is valid )

proof end;

theorem :: CQC_THE2:89

for A being QC-alphabet

for p, q, r being Element of CQC-WFF A holds

( {p,q} |- r iff {(p '&' q)} |- r )

for p, q, r being Element of CQC-WFF A holds

( {p,q} |- r iff {(p '&' q)} |- r )

proof end;

theorem Th90: :: CQC_THE2:90

for A being QC-alphabet

for X being Subset of (CQC-WFF A)

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st X |- p holds

X |- All (x,p)

for X being Subset of (CQC-WFF A)

for p being Element of CQC-WFF A

for x being bound_QC-variable of A st X |- p holds

X |- All (x,p)

proof end;

theorem :: CQC_THE2:91

for A being QC-alphabet

for X being Subset of (CQC-WFF A)

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

X |- (All (x,(p => q))) => (p => (All (x,q))) by Th74, CQC_THE1:59;

for X being Subset of (CQC-WFF A)

for p, q being Element of CQC-WFF A

for x being bound_QC-variable of A st not x in still_not-bound_in p holds

X |- (All (x,(p => q))) => (p => (All (x,q))) by Th74, CQC_THE1:59;

theorem :: CQC_THE2:92

for A being QC-alphabet

for X being Subset of (CQC-WFF A)

for F, G being Element of CQC-WFF A st F is closed & X \/ {F} |- G holds

X |- F => G

for X being Subset of (CQC-WFF A)

for F, G being Element of CQC-WFF A st F is closed & X \/ {F} |- G holds

X |- F => G

proof end;