Article CQC_THE3, MML version 4.99.1005

:: CQC_THE3:th 1
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bool CQC-WFF
      st b1 in b2
   holds b2 |- b1;

:: CQC_THE3:th 2
theorem
for b1, b2 being Element of bool CQC-WFF
      st b1 c= Cn b2
   holds Cn b1 c= Cn b2;

:: CQC_THE3:th 3
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
      st b3 |- b1 & {b1} |- b2
   holds b3 |- b2;

:: CQC_THE3:th 4
theorem
for b1 being Element of CQC-WFF
for b2, b3 being Element of bool CQC-WFF
      st b2 |- b1 & b2 c= b3
   holds b3 |- b1;

:: CQC_THE3:prednot 1 => CQC_THE3:pred 1
definition
  let a1, a2 be Element of CQC-WFF;
  pred A1 |- A2 means
    {a1} |- a2;
end;

:: CQC_THE3:dfs 1
definiens
  let a1, a2 be Element of CQC-WFF;
To prove
     a1 |- a2
it is sufficient to prove
  thus {a1} |- a2;

:: CQC_THE3:def 1
theorem
for b1, b2 being Element of CQC-WFF holds
   b1 |- b2
iff
   {b1} |- b2;

:: CQC_THE3:th 5
theorem
for b1 being Element of CQC-WFF holds
   b1 |- b1;

:: CQC_THE3:th 6
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 |- b2 & b2 |- b3
   holds b1 |- b3;

:: CQC_THE3:prednot 2 => CQC_THE3:pred 2
definition
  let a1, a2 be Element of bool CQC-WFF;
  pred A1 |- A2 means
    for b1 being Element of CQC-WFF
          st b1 in a2
       holds a1 |- b1;
end;

:: CQC_THE3:dfs 2
definiens
  let a1, a2 be Element of bool CQC-WFF;
To prove
     a1 |- a2
it is sufficient to prove
  thus for b1 being Element of CQC-WFF
          st b1 in a2
       holds a1 |- b1;

:: CQC_THE3:def 2
theorem
for b1, b2 being Element of bool CQC-WFF holds
   b1 |- b2
iff
   for b3 being Element of CQC-WFF
         st b3 in b2
      holds b1 |- b3;

:: CQC_THE3:th 7
theorem
for b1, b2 being Element of bool CQC-WFF holds
   b1 |- b2
iff
   b2 c= Cn b1;

:: CQC_THE3:th 8
theorem
for b1 being Element of bool CQC-WFF holds
   b1 |- b1;

:: CQC_THE3:th 9
theorem
for b1, b2, b3 being Element of bool CQC-WFF
      st b1 |- b2 & b2 |- b3
   holds b1 |- b3;

:: CQC_THE3:th 10
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bool CQC-WFF holds
      b2 |- {b1}
   iff
      b2 |- b1;

:: CQC_THE3:th 11
theorem
for b1, b2 being Element of CQC-WFF holds
   {b1} |- {b2}
iff
   b1 |- b2;

:: CQC_THE3:th 12
theorem
for b1, b2 being Element of bool CQC-WFF
      st b1 c= b2
   holds b2 |- b1;

:: CQC_THE3:th 13
theorem
for b1 being Element of bool CQC-WFF holds
   b1 |- TAUT;

:: CQC_THE3:th 14
theorem
{} CQC-WFF |- TAUT;

:: CQC_THE3:prednot 3 => CQC_THE3:pred 3
definition
  let a1 be Element of bool CQC-WFF;
  pred |- A1 means
    for b1 being Element of CQC-WFF
          st b1 in a1
       holds b1 is valid;
end;

:: CQC_THE3:dfs 3
definiens
  let a1 be Element of bool CQC-WFF;
To prove
     |- a1
it is sufficient to prove
  thus for b1 being Element of CQC-WFF
          st b1 in a1
       holds b1 is valid;

:: CQC_THE3:def 3
theorem
for b1 being Element of bool CQC-WFF holds
      |- b1
   iff
      for b2 being Element of CQC-WFF
            st b2 in b1
         holds b2 is valid;

:: CQC_THE3:th 15
theorem
for b1 being Element of bool CQC-WFF holds
      |- b1
   iff
      {} CQC-WFF |- b1;

:: CQC_THE3:th 16
theorem
|- TAUT;

:: CQC_THE3:th 17
theorem
for b1 being Element of bool CQC-WFF holds
      |- b1
   iff
      b1 c= TAUT;

:: CQC_THE3:prednot 4 => CQC_THE3:pred 4
definition
  let a1, a2 be Element of bool CQC-WFF;
  pred A1 |-| A2 means
    for b1 being Element of CQC-WFF holds
          a1 |- b1
       iff
          a2 |- b1;
  symmetry;
::  for a1, a2 being Element of bool CQC-WFF
::        st a1 |-| a2
::     holds a2 |-| a1;
  reflexivity;
::  for a1 being Element of bool CQC-WFF holds
::     a1 |-| a1;
end;

:: CQC_THE3:dfs 4
definiens
  let a1, a2 be Element of bool CQC-WFF;
To prove
     a1 |-| a2
it is sufficient to prove
  thus for b1 being Element of CQC-WFF holds
          a1 |- b1
       iff
          a2 |- b1;

:: CQC_THE3:def 4
theorem
for b1, b2 being Element of bool CQC-WFF holds
   b1 |-| b2
iff
   for b3 being Element of CQC-WFF holds
         b1 |- b3
      iff
         b2 |- b3;

:: CQC_THE3:th 18
theorem
for b1, b2 being Element of bool CQC-WFF holds
   b1 |-| b2
iff
   b1 |- b2 & b2 |- b1;

:: CQC_THE3:th 19
theorem
for b1, b2, b3 being Element of bool CQC-WFF
      st b1 |-| b2 & b2 |-| b3
   holds b1 |-| b3;

:: CQC_THE3:th 20
theorem
for b1, b2 being Element of bool CQC-WFF holds
   b1 |-| b2
iff
   Cn b1 = Cn b2;

:: CQC_THE3:th 21
theorem
for b1, b2 being Element of bool CQC-WFF holds
(Cn b1) \/ Cn b2 c= Cn (b1 \/ b2);

:: CQC_THE3:th 22
theorem
for b1, b2 being Element of bool CQC-WFF holds
Cn (b1 \/ b2) = Cn ((Cn b1) \/ Cn b2);

:: CQC_THE3:th 23
theorem
for b1 being Element of bool CQC-WFF holds
   b1 |-| Cn b1;

:: CQC_THE3:th 24
theorem
for b1, b2 being Element of bool CQC-WFF holds
b1 \/ b2 |-| (Cn b1) \/ Cn b2;

:: CQC_THE3:th 25
theorem
for b1, b2, b3 being Element of bool CQC-WFF
      st b1 |-| b2
   holds b1 \/ b3 |-| b2 \/ b3;

:: CQC_THE3:th 26
theorem
for b1, b2, b3, b4 being Element of bool CQC-WFF
      st b1 |-| b2 & b1 \/ b3 |- b4
   holds b2 \/ b3 |- b4;

:: CQC_THE3:th 27
theorem
for b1, b2, b3 being Element of bool CQC-WFF
      st b1 |-| b2 & b3 |- b1
   holds b3 |- b2;

:: CQC_THE3:prednot 5 => CQC_THE3:pred 5
definition
  let a1, a2 be Element of CQC-WFF;
  pred A1 |-| A2 means
    a1 |- a2 & a2 |- a1;
  symmetry;
::  for a1, a2 being Element of CQC-WFF
::        st a1 |-| a2
::     holds a2 |-| a1;
  reflexivity;
::  for a1 being Element of CQC-WFF holds
::     a1 |-| a1;
end;

:: CQC_THE3:dfs 5
definiens
  let a1, a2 be Element of CQC-WFF;
To prove
     a1 |-| a2
it is sufficient to prove
  thus a1 |- a2 & a2 |- a1;

:: CQC_THE3:def 5
theorem
for b1, b2 being Element of CQC-WFF holds
   b1 |-| b2
iff
   b1 |- b2 & b2 |- b1;

:: CQC_THE3:th 28
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 |-| b2 & b2 |-| b3
   holds b1 |-| b3;

:: CQC_THE3:th 29
theorem
for b1, b2 being Element of CQC-WFF holds
   b1 |-| b2
iff
   {b1} |-| {b2};

:: CQC_THE3:th 30
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
      st b1 |-| b2 & b3 |- b1
   holds b3 |- b2;

:: CQC_THE3:th 31
theorem
for b1, b2 being Element of CQC-WFF holds
{b1,b2} |-| {b1 '&' b2};

:: CQC_THE3:th 32
theorem
for b1, b2 being Element of CQC-WFF holds
b1 '&' b2 |-| b2 '&' b1;

:: CQC_THE3:th 33
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF holds
      b3 |- b1 '&' b2
   iff
      b3 |- b1 & b3 |- b2;

:: CQC_THE3:th 34
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 |-| b2 & b3 |-| b4
   holds b1 '&' b3 |-| b2 '&' b4;

:: CQC_THE3:th 35
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bool CQC-WFF
for b3 being Element of bound_QC-variables holds
      b2 |- All(b3,b1)
   iff
      b2 |- b1;

:: CQC_THE3:th 36
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bound_QC-variables holds
   All(b2,b1) |-| b1;

:: CQC_THE3:th 37
theorem
for b1, b2 being Element of CQC-WFF
for b3, b4 being Element of bound_QC-variables
      st b1 |-| b2
   holds All(b3,b1) |-| All(b4,b2);

:: CQC_THE3:prednot 6 => CQC_THE3:pred 6
definition
  let a1, a2 be Element of CQC-WFF;
  pred A1 is_an_universal_closure_of A2 means
    a1 is closed &
     (ex b1 being Element of NAT st
        1 <= b1 &
         (ex b2 being Relation-like Function-like FinSequence-like set st
            len b2 = b1 &
             b2 . 1 = a2 &
             b2 . b1 = a1 &
             (for b3 being Element of NAT
                   st 1 <= b3 & b3 < b1
                holds ex b4 being Element of bound_QC-variables st
                   ex b5 being Element of CQC-WFF st
                      b5 = b2 . b3 & b2 . (b3 + 1) = All(b4,b5))));
end;

:: CQC_THE3:dfs 6
definiens
  let a1, a2 be Element of CQC-WFF;
To prove
     a1 is_an_universal_closure_of a2
it is sufficient to prove
  thus a1 is closed &
     (ex b1 being Element of NAT st
        1 <= b1 &
         (ex b2 being Relation-like Function-like FinSequence-like set st
            len b2 = b1 &
             b2 . 1 = a2 &
             b2 . b1 = a1 &
             (for b3 being Element of NAT
                   st 1 <= b3 & b3 < b1
                holds ex b4 being Element of bound_QC-variables st
                   ex b5 being Element of CQC-WFF st
                      b5 = b2 . b3 & b2 . (b3 + 1) = All(b4,b5))));

:: CQC_THE3:def 6
theorem
for b1, b2 being Element of CQC-WFF holds
   b1 is_an_universal_closure_of b2
iff
   b1 is closed &
    (ex b3 being Element of NAT st
       1 <= b3 &
        (ex b4 being Relation-like Function-like FinSequence-like set st
           len b4 = b3 &
            b4 . 1 = b2 &
            b4 . b3 = b1 &
            (for b5 being Element of NAT
                  st 1 <= b5 & b5 < b3
               holds ex b6 being Element of bound_QC-variables st
                  ex b7 being Element of CQC-WFF st
                     b7 = b4 . b5 & b4 . (b5 + 1) = All(b6,b7))));

:: CQC_THE3:th 38
theorem
for b1, b2 being Element of CQC-WFF
      st b1 is_an_universal_closure_of b2
   holds b1 |-| b2;

:: CQC_THE3:th 39
theorem
for b1, b2 being Element of CQC-WFF
      st b1 => b2 is valid
   holds b1 |- b2;

:: CQC_THE3:th 40
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
      st b3 |- b1 => b2
   holds b3 \/ {b1} |- b2;

:: CQC_THE3:th 41
theorem
for b1, b2 being Element of CQC-WFF
      st b1 is closed & b1 |- b2
   holds b1 => b2 is valid;

:: CQC_THE3:th 42
theorem
for b1, b2, b3 being Element of CQC-WFF
for b4 being Element of bool CQC-WFF
      st b1 is_an_universal_closure_of b2
   holds    b4 \/ {b2} |- b3
   iff
      b4 |- b1 => b3;

:: CQC_THE3:th 43
theorem
for b1, b2 being Element of CQC-WFF
      st b1 is closed & b1 |- b2
   holds 'not' b2 |- 'not' b1;

:: CQC_THE3:th 44
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
      st b1 is closed & b3 \/ {b1} |- b2
   holds b3 \/ {'not' b2} |- 'not' b1;

:: CQC_THE3:th 45
theorem
for b1, b2 being Element of CQC-WFF
      st b1 is closed & 'not' b1 |- 'not' b2
   holds b2 |- b1;

:: CQC_THE3:th 46
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
      st b1 is closed &
         b3 \/ {'not' b1} |- 'not' b2
   holds b3 \/ {b2} |- b1;

:: CQC_THE3:th 47
theorem
for b1, b2 being Element of CQC-WFF
      st b1 is closed & b2 is closed
   holds    b1 |- b2
   iff
      'not' b2 |- 'not' b1;

:: CQC_THE3:th 48
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 is_an_universal_closure_of b2 & b3 is_an_universal_closure_of b4
   holds    b2 |- b4
   iff
      'not' b3 |- 'not' b1;

:: CQC_THE3:th 49
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 is_an_universal_closure_of b2 & b3 is_an_universal_closure_of b4
   holds    b2 |-| b4
   iff
      'not' b1 |-| 'not' b3;

:: CQC_THE3:prednot 7 => CQC_THE3:pred 7
definition
  let a1, a2 be Element of CQC-WFF;
  pred A1 <==> A2 means
    a1 <=> a2 is valid;
  symmetry;
::  for a1, a2 being Element of CQC-WFF
::        st a1 <==> a2
::     holds a2 <==> a1;
  reflexivity;
::  for a1 being Element of CQC-WFF holds
::     a1 <==> a1;
end;

:: CQC_THE3:dfs 7
definiens
  let a1, a2 be Element of CQC-WFF;
To prove
     a1 <==> a2
it is sufficient to prove
  thus a1 <=> a2 is valid;

:: CQC_THE3:def 7
theorem
for b1, b2 being Element of CQC-WFF holds
   b1 <==> b2
iff
   b1 <=> b2 is valid;

:: CQC_THE3:th 50
theorem
for b1, b2 being Element of CQC-WFF holds
   b1 <==> b2
iff
   b1 => b2 is valid & b2 => b1 is valid;

:: CQC_THE3:th 51
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 <==> b2 & b2 <==> b3
   holds b1 <==> b3;

:: CQC_THE3:th 52
theorem
for b1, b2 being Element of CQC-WFF
      st b1 <==> b2
   holds b1 |-| b2;

:: CQC_THE3:th 53
theorem
for b1, b2 being Element of CQC-WFF holds
   b1 <==> b2
iff
   'not' b1 <==> 'not' b2;

:: CQC_THE3:th 54
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 <==> b2 & b3 <==> b4
   holds b1 '&' b3 <==> b2 '&' b4;

:: CQC_THE3:th 55
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 <==> b2 & b3 <==> b4
   holds b1 => b3 <==> b2 => b4;

:: CQC_THE3:th 56
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 <==> b2 & b3 <==> b4
   holds b1 'or' b3 <==> b2 'or' b4;

:: CQC_THE3:th 57
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 <==> b2 & b3 <==> b4
   holds b1 <=> b3 <==> b2 <=> b4;

:: CQC_THE3:th 58
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st b1 <==> b2
   holds All(b3,b1) <==> All(b3,b2);

:: CQC_THE3:th 59
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bound_QC-variables
      st b1 <==> b2
   holds Ex(b3,b1) <==> Ex(b3,b2);

:: CQC_THE3:th 61
theorem
for b1 being Element of NAT
for b2 being QC-variable_list of b1
for b3 being Element of free_QC-variables
for b4 being Element of bound_QC-variables holds
   still_not-bound_in b2 c= still_not-bound_in Subst(b2,b3 .--> b4);

:: CQC_THE3:th 62
theorem
for b1 being Element of NAT
for b2 being QC-variable_list of b1
for b3 being Element of free_QC-variables
for b4 being Element of bound_QC-variables holds
   still_not-bound_in Subst(b2,b3 .--> b4) c= (still_not-bound_in b2) \/ {b4};

:: CQC_THE3:th 63
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of QC-WFF holds
   still_not-bound_in b2 c= still_not-bound_in (b2 . b1);

:: CQC_THE3:th 64
theorem
for b1 being Element of bound_QC-variables
for b2 being Element of QC-WFF holds
   still_not-bound_in (b2 . b1) c= (still_not-bound_in b2) \/ {b1};

:: CQC_THE3:th 65
theorem
for b1 being Element of CQC-WFF
for b2 being Element of QC-WFF
for b3, b4 being Element of bound_QC-variables
      st b1 = b2 . b3 & b3 <> b4 & not b4 in still_not-bound_in b2
   holds not b4 in still_not-bound_in b1;

:: CQC_THE3:th 66
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 b3
   holds All(b4,b1) <==> All(b5,b2);