Article CQC_THE2, MML version 4.99.1005

:: CQC_THE2:th 1
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 => (b2 => b3) is valid
   holds (b1 '&' b2) => b3 is valid;

:: CQC_THE2:th 2
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 => (b2 => b3) is valid
   holds (b2 '&' b1) => b3 is valid;

:: CQC_THE2:th 3
theorem
for b1, b2, b3 being Element of CQC-WFF
      st (b1 '&' b2) => b3 is valid
   holds b1 => (b2 => b3) is valid;

:: CQC_THE2:th 4
theorem
for b1, b2, b3 being Element of CQC-WFF
      st (b1 '&' b2) => b3 is valid
   holds b2 => (b1 => b3) is valid;

:: CQC_THE2:th 5
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of bound_QC-variables holds
   b2 in still_not-bound_in All(b3,b1)
iff
   b2 in still_not-bound_in b1 & b2 <> b3;

:: CQC_THE2:th 6
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of bound_QC-variables holds
   b2 in still_not-bound_in Ex(b3,b1)
iff
   b2 in still_not-bound_in b1 & b2 <> b3;

:: CQC_THE2:th 7
theorem
for b1, b2 being Element of QC-WFF
for b3 being Element of bound_QC-variables holds
      b3 in still_not-bound_in (b1 => b2)
   iff
      (b3 in still_not-bound_in b1 or b3 in still_not-bound_in b2);

:: CQC_THE2:th 9
theorem
for b1, b2 being Element of QC-WFF
for b3 being Element of bound_QC-variables holds
      b3 in still_not-bound_in (b1 '&' b2)
   iff
      (b3 in still_not-bound_in b1 or b3 in still_not-bound_in b2);

:: CQC_THE2:th 10
theorem
for b1, b2 being Element of QC-WFF
for b3 being Element of bound_QC-variables holds
      b3 in still_not-bound_in (b1 'or' b2)
   iff
      (b3 in still_not-bound_in b1 or b3 in still_not-bound_in b2);

:: CQC_THE2:th 11
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of bound_QC-variables holds
not b2 in still_not-bound_in All(b2,b3,b1) & not b3 in still_not-bound_in All(b2,b3,b1);

:: CQC_THE2:th 12
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of bound_QC-variables holds
not b2 in still_not-bound_in Ex(b2,b3,b1) & not b3 in still_not-bound_in Ex(b2,b3,b1);

:: CQC_THE2:th 14
theorem
for b1, b2 being Element of QC-WFF
for b3 being Element of bound_QC-variables holds
   (b1 => b2) . b3 = (b1 . b3) => (b2 . b3);

:: CQC_THE2:th 15
theorem
for b1, b2 being Element of QC-WFF
for b3 being Element of bound_QC-variables holds
   (b1 'or' b2) . b3 = (b1 . b3) 'or' (b2 . b3);

:: CQC_THE2:th 17
theorem
for b1 being Element of CQC-WFF
for b2, b3 being Element of bound_QC-variables
      st b2 <> b3
   holds (Ex(b2,b1)) . b3 = Ex(b2,b1 . b3);

:: CQC_THE2:th 18
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   b1 => Ex(b2,b1) is valid;

:: CQC_THE2:th 19
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
      st b1 is valid
   holds Ex(b2,b1) is valid;

:: CQC_THE2:th 20
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   (All(b2,b1)) => Ex(b2,b1) is valid;

:: CQC_THE2:th 21
theorem
for b1 being Element of CQC-WFF
for b2, b3 being Element of bound_QC-variables holds
(All(b2,b1)) => Ex(b3,b1) is valid;

:: CQC_THE2:th 22
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st b1 => b2 is valid & not b3 in still_not-bound_in b2
   holds (Ex(b3,b1)) => b2 is valid;

:: CQC_THE2:th 23
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
      st not b2 in still_not-bound_in b1
   holds (Ex(b2,b1)) => b1 is valid;

:: CQC_THE2:th 24
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
      st not b2 in still_not-bound_in b1 & Ex(b2,b1) is valid
   holds b1 is valid;

:: CQC_THE2:th 25
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of QC-WFF
for b4, b5 being Element of bound_QC-variables
      st b1 = b3 . b4 & b2 = b3 . b5 & not b5 in still_not-bound_in b3
   holds b1 => Ex(b5,b2) is valid;

:: CQC_THE2:th 26
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
      st b1 is valid
   holds All(b2,b1) is valid;

:: CQC_THE2:th 27
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables
      st not b2 in still_not-bound_in b1
   holds b1 => All(b2,b1) is valid;

:: CQC_THE2:th 28
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of QC-WFF
for b4, b5 being Element of bound_QC-variables
      st b1 = b3 . b4 & b2 = b3 . b5 & not b4 in still_not-bound_in b3
   holds (All(b4,b1)) => b2 is valid;

:: CQC_THE2:th 29
theorem
for b1 being Element of CQC-WFF
for b2, b3 being Element of bound_QC-variables
      st not b2 in still_not-bound_in b1
   holds (All(b3,b1)) => All(b2,b1) is valid;

:: CQC_THE2:th 30
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of QC-WFF
for b4, b5 being Element of bound_QC-variables
      st b1 = b3 . b4 & b2 = b3 . b5 & not b4 in still_not-bound_in b3 & not b5 in still_not-bound_in b1
   holds (All(b4,b1)) => All(b5,b2) is valid;

:: CQC_THE2:th 31
theorem
for b1 being Element of CQC-WFF
for b2, b3 being Element of bound_QC-variables
      st not b2 in still_not-bound_in b1
   holds (Ex(b2,b1)) => Ex(b3,b1) is valid;

:: CQC_THE2:th 32
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of QC-WFF
for b4, b5 being Element of bound_QC-variables
      st b1 = b3 . b4 & b2 = b3 . b5 & not b4 in still_not-bound_in b2 & not b5 in still_not-bound_in b3
   holds (Ex(b4,b1)) => Ex(b5,b2) is valid;

:: CQC_THE2:th 34
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (All(b3,b1 => b2)) => ((All(b3,b1)) => All(b3,b2)) is valid;

:: CQC_THE2:th 35
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st All(b3,b1 => b2) is valid
   holds (All(b3,b1)) => All(b3,b2) is valid;

:: CQC_THE2:th 36
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (All(b3,b1 <=> b2)) => ((All(b3,b1)) <=> All(b3,b2)) is valid;

:: CQC_THE2:th 37
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st All(b3,b1 <=> b2) is valid
   holds (All(b3,b1)) <=> All(b3,b2) is valid;

:: CQC_THE2:th 38
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (All(b3,b1 => b2)) => ((Ex(b3,b1)) => Ex(b3,b2)) is valid;

:: CQC_THE2:th 39
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st All(b3,b1 => b2) is valid
   holds (Ex(b3,b1)) => Ex(b3,b2) is valid;

:: CQC_THE2:th 40
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (All(b3,b1 '&' b2)) => ((All(b3,b1)) '&' All(b3,b2)) is valid &
    ((All(b3,b1)) '&' All(b3,b2)) => All(b3,b1 '&' b2) is valid;

:: CQC_THE2:th 41
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (All(b3,b1 '&' b2)) <=> ((All(b3,b1)) '&' All(b3,b2)) is valid;

:: CQC_THE2:th 42
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
      All(b3,b1 '&' b2) is valid
   iff
      (All(b3,b1)) '&' All(b3,b2) is valid;

:: CQC_THE2:th 43
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   ((All(b3,b1)) 'or' All(b3,b2)) => All(b3,b1 'or' b2) is valid;

:: CQC_THE2:th 44
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (Ex(b3,b1 'or' b2)) => ((Ex(b3,b1)) 'or' Ex(b3,b2)) is valid &
    ((Ex(b3,b1)) 'or' Ex(b3,b2)) => Ex(b3,b1 'or' b2) is valid;

:: CQC_THE2:th 45
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (Ex(b3,b1 'or' b2)) <=> ((Ex(b3,b1)) 'or' Ex(b3,b2)) is valid;

:: CQC_THE2:th 46
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
      Ex(b3,b1 'or' b2) is valid
   iff
      (Ex(b3,b1)) 'or' Ex(b3,b2) is valid;

:: CQC_THE2:th 47
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (Ex(b3,b1 '&' b2)) => ((Ex(b3,b1)) '&' Ex(b3,b2)) is valid;

:: CQC_THE2:th 48
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st Ex(b3,b1 '&' b2) is valid
   holds (Ex(b3,b1)) '&' Ex(b3,b2) is valid;

:: CQC_THE2:th 49
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   (All(b2,'not' 'not' b1)) => All(b2,b1) is valid &
    (All(b2,b1)) => All(b2,'not' 'not' b1) is valid;

:: CQC_THE2:th 50
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   (All(b2,'not' 'not' b1)) <=> All(b2,b1) is valid;

:: CQC_THE2:th 51
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   (Ex(b2,'not' 'not' b1)) => Ex(b2,b1) is valid &
    (Ex(b2,b1)) => Ex(b2,'not' 'not' b1) is valid;

:: CQC_THE2:th 52
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   (Ex(b2,'not' 'not' b1)) <=> Ex(b2,b1) is valid;

:: CQC_THE2:th 53
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   ('not' Ex(b2,'not' b1)) => All(b2,b1) is valid &
    (All(b2,b1)) => 'not' Ex(b2,'not' b1) is valid;

:: CQC_THE2:th 54
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   ('not' Ex(b2,'not' b1)) <=> All(b2,b1) is valid;

:: CQC_THE2:th 55
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   ('not' All(b2,b1)) => Ex(b2,'not' b1) is valid &
    (Ex(b2,'not' b1)) => 'not' All(b2,b1) is valid;

:: CQC_THE2:th 56
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   ('not' All(b2,b1)) <=> Ex(b2,'not' b1) is valid;

:: CQC_THE2:th 57
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   ('not' Ex(b2,b1)) => All(b2,'not' b1) is valid &
    (All(b2,'not' b1)) => 'not' Ex(b2,b1) is valid;

:: CQC_THE2:th 58
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   (All(b2,'not' b1)) <=> 'not' Ex(b2,b1) is valid;

:: CQC_THE2:th 59
theorem
for b1 being Element of CQC-WFF
for b2, b3 being Element of bound_QC-variables holds
(All(b2,All(b3,b1))) => All(b3,All(b2,b1)) is valid &
 (All(b2,b3,b1)) => All(b3,b2,b1) is valid;

:: CQC_THE2:th 60
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of QC-WFF
for b4, b5 being Element of bound_QC-variables
      st b1 = b3 . b4 & b2 = b3 . b5 & not b5 in still_not-bound_in b3
   holds (All(b4,All(b5,b2))) => All(b4,b1) is valid;

:: CQC_THE2:th 61
theorem
for b1 being Element of CQC-WFF
for b2, b3 being Element of bound_QC-variables holds
(Ex(b2,Ex(b3,b1))) => Ex(b3,Ex(b2,b1)) is valid &
 (Ex(b2,b3,b1)) => Ex(b3,b2,b1) is valid;

:: CQC_THE2:th 62
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of QC-WFF
for b4, b5 being Element of bound_QC-variables
      st b1 = b3 . b4 & b2 = b3 . b5 & not b5 in still_not-bound_in b3
   holds (Ex(b4,b1)) => Ex(b4,b5,b2) is valid;

:: CQC_THE2:th 63
theorem
for b1 being Element of CQC-WFF
for b2, b3 being Element of bound_QC-variables holds
(Ex(b2,All(b3,b1))) => All(b3,Ex(b2,b1)) is valid;

:: CQC_THE2:th 64
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   Ex(b2,b1 <=> b1) is valid;

:: CQC_THE2:th 65
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (Ex(b3,b1 => b2)) => ((All(b3,b1)) => Ex(b3,b2)) is valid &
    ((All(b3,b1)) => Ex(b3,b2)) => Ex(b3,b1 => b2) is valid;

:: CQC_THE2:th 66
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (Ex(b3,b1 => b2)) <=> ((All(b3,b1)) => Ex(b3,b2)) is valid;

:: CQC_THE2:th 67
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
      Ex(b3,b1 => b2) is valid
   iff
      (All(b3,b1)) => Ex(b3,b2) is valid;

:: CQC_THE2:th 68
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (All(b3,b1 '&' b2)) => (b1 '&' All(b3,b2)) is valid;

:: CQC_THE2:th 69
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (All(b3,b1 '&' b2)) => ((All(b3,b1)) '&' b2) is valid;

:: CQC_THE2:th 70
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (b1 '&' All(b3,b2)) => All(b3,b1 '&' b2) is valid;

:: CQC_THE2:th 71
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1 & b1 '&' All(b3,b2) is valid
   holds All(b3,b1 '&' b2) is valid;

:: CQC_THE2:th 72
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (b1 'or' All(b3,b2)) => All(b3,b1 'or' b2) is valid &
    (All(b3,b1 'or' b2)) => (b1 'or' All(b3,b2)) is valid;

:: CQC_THE2:th 73
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (b1 'or' All(b3,b2)) <=> All(b3,b1 'or' b2) is valid;

:: CQC_THE2:th 74
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds    b1 'or' All(b3,b2) is valid
   iff
      All(b3,b1 'or' b2) is valid;

:: CQC_THE2:th 75
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (b1 '&' Ex(b3,b2)) => Ex(b3,b1 '&' b2) is valid &
    (Ex(b3,b1 '&' b2)) => (b1 '&' Ex(b3,b2)) is valid;

:: CQC_THE2:th 76
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (b1 '&' Ex(b3,b2)) <=> Ex(b3,b1 '&' b2) is valid;

:: CQC_THE2:th 77
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds    b1 '&' Ex(b3,b2) is valid
   iff
      Ex(b3,b1 '&' b2) is valid;

:: CQC_THE2:th 78
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (All(b3,b1 => b2)) => (b1 => All(b3,b2)) is valid &
    (b1 => All(b3,b2)) => All(b3,b1 => b2) is valid;

:: CQC_THE2:th 79
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (b1 => All(b3,b2)) <=> All(b3,b1 => b2) is valid;

:: CQC_THE2:th 80
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds    All(b3,b1 => b2) is valid
   iff
      b1 => All(b3,b2) is valid;

:: CQC_THE2:th 81
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (Ex(b3,b2 => b1)) => ((All(b3,b2)) => b1) is valid;

:: CQC_THE2:th 82
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   ((All(b3,b1)) => b2) => Ex(b3,b1 => b2) is valid;

:: CQC_THE2:th 83
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds    (All(b3,b2)) => b1 is valid
   iff
      Ex(b3,b2 => b1) is valid;

:: CQC_THE2:th 84
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds ((Ex(b3,b2)) => b1) => All(b3,b2 => b1) is valid &
    (All(b3,b2 => b1)) => ((Ex(b3,b2)) => b1) is valid;

:: CQC_THE2:th 85
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds ((Ex(b3,b2)) => b1) <=> All(b3,b2 => b1) is valid;

:: CQC_THE2:th 86
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds    (Ex(b3,b2)) => b1 is valid
   iff
      All(b3,b2 => b1) is valid;

:: CQC_THE2:th 87
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (Ex(b3,b1 => b2)) => (b1 => Ex(b3,b2)) is valid;

:: CQC_THE2:th 88
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables holds
   (b1 => Ex(b3,b2)) => Ex(b3,b1 => b2) is valid;

:: CQC_THE2:th 89
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds (b1 => Ex(b3,b2)) <=> Ex(b3,b1 => b2) is valid;

:: CQC_THE2:th 90
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st not b3 in still_not-bound_in b1
   holds    b1 => Ex(b3,b2) is valid
   iff
      Ex(b3,b1 => b2) is valid;

:: CQC_THE2:th 91
theorem
for b1 being Element of CQC-WFF holds
   {b1} |- b1;

:: CQC_THE2:th 92
theorem
for b1, b2 being Element of CQC-WFF holds
Cn ({b1} \/ {b2}) = Cn {b1 '&' b2};

:: CQC_THE2:th 93
theorem
for b1, b2, b3 being Element of CQC-WFF holds
   {b1,b2} |- b3
iff
   {b1 '&' b2} |- b3;

:: CQC_THE2:th 94
theorem
for b1 being Element of bool CQC-WFF
for b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st b1 |- b2
   holds b1 |- All(b3,b2);

:: CQC_THE2:th 95
theorem
for b1 being Element of bool CQC-WFF
for b2, b3 being Element of CQC-WFF
for b4 being Element of bound_QC-variables
      st not b4 in still_not-bound_in b2
   holds b1 |- (All(b4,b2 => b3)) => (b2 => All(b4,b3));

:: CQC_THE2:th 96
theorem
for b1 being Element of bool CQC-WFF
for b2, b3 being Element of CQC-WFF
      st b2 is closed & b1 \/ {b2} |- b3
   holds b1 |- b2 => b3;