Article PROCAL_1, MML version 4.99.1005

:: PROCAL_1:th 1
theorem
for b1 being Element of CQC-WFF holds
   'not' (b1 '&' 'not' b1) in TAUT;

:: PROCAL_1:th 2
theorem
for b1 being Element of CQC-WFF holds
   b1 'or' 'not' b1 in TAUT;

:: PROCAL_1:th 3
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => (b1 'or' b2) in TAUT;

:: PROCAL_1:th 4
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => (b2 'or' b1) in TAUT;

:: PROCAL_1:th 5
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 'or' b2) => (('not' b1) => b2) in TAUT;

:: PROCAL_1:th 6
theorem
for b1, b2 being Element of CQC-WFF holds
('not' (b1 'or' b2)) => (('not' b1) '&' 'not' b2) in TAUT;

:: PROCAL_1:th 7
theorem
for b1, b2 being Element of CQC-WFF holds
(('not' b1) '&' 'not' b2) => 'not' (b1 'or' b2) in TAUT;

:: PROCAL_1:th 8
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 'or' b2) => (b2 'or' b1) in TAUT;

:: PROCAL_1:th 9
theorem
for b1 being Element of CQC-WFF holds
   ('not' b1) 'or' b1 in TAUT;

:: PROCAL_1:th 10
theorem
for b1, b2 being Element of CQC-WFF holds
('not' (b1 'or' b2)) => 'not' b1 in TAUT;

:: PROCAL_1:th 11
theorem
for b1 being Element of CQC-WFF holds
   (b1 'or' b1) => b1 in TAUT;

:: PROCAL_1:th 12
theorem
for b1 being Element of CQC-WFF holds
   b1 => (b1 'or' b1) in TAUT;

:: PROCAL_1:th 13
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 '&' 'not' b1) => b2 in TAUT;

:: PROCAL_1:th 14
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => b2) => (('not' b1) 'or' b2) in TAUT;

:: PROCAL_1:th 15
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 '&' b2) => 'not' (b1 => 'not' b2) in TAUT;

:: PROCAL_1:th 16
theorem
for b1, b2 being Element of CQC-WFF holds
('not' (b1 => 'not' b2)) => (b1 '&' b2) in TAUT;

:: PROCAL_1:th 17
theorem
for b1, b2 being Element of CQC-WFF holds
('not' (b1 '&' b2)) => (('not' b1) 'or' 'not' b2) in TAUT;

:: PROCAL_1:th 18
theorem
for b1, b2 being Element of CQC-WFF holds
(('not' b1) 'or' 'not' b2) => 'not' (b1 '&' b2) in TAUT;

:: PROCAL_1:th 19
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 '&' b2) => b1 in TAUT;

:: PROCAL_1:th 20
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 '&' b2) => (b1 'or' b2) in TAUT;

:: PROCAL_1:th 21
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 '&' b2) => b2 in TAUT;

:: PROCAL_1:th 22
theorem
for b1 being Element of CQC-WFF holds
   b1 => (b1 '&' b1) in TAUT;

:: PROCAL_1:th 23
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 <=> b2) => (b1 => b2) in TAUT;

:: PROCAL_1:th 24
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 <=> b2) => (b2 => b1) in TAUT;

:: PROCAL_1:th 25
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 'or' b2) 'or' b3) => (b1 'or' (b2 'or' b3)) in TAUT;

:: PROCAL_1:th 26
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 '&' b2) '&' b3) => (b1 '&' (b2 '&' b3)) in TAUT;

:: PROCAL_1:th 27
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 'or' (b2 'or' b3)) => ((b1 'or' b2) 'or' b3) in TAUT;

:: PROCAL_1:th 28
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => (b2 => (b1 '&' b2)) in TAUT;

:: PROCAL_1:th 29
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => b2) => ((b2 => b1) => (b1 <=> b2)) in TAUT;

:: PROCAL_1:th 30
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 'or' b2) <=> (b2 'or' b1) in TAUT;

:: PROCAL_1:th 31
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 '&' b2) => b3) => (b1 => (b2 => b3)) in TAUT;

:: PROCAL_1:th 32
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => (b2 => b3)) => ((b1 '&' b2) => b3) in TAUT;

:: PROCAL_1:th 33
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => b2) => ((b1 => b3) => (b1 => (b2 '&' b3))) in TAUT;

:: PROCAL_1:th 34
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 'or' b2) => b3) => ((b1 => b3) 'or' (b2 => b3)) in TAUT;

:: PROCAL_1:th 35
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => b2) => ((b3 => b2) => ((b1 'or' b3) => b2)) in TAUT;

:: PROCAL_1:th 36
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 => b2) '&' (b3 => b2)) => ((b1 'or' b3) => b2) in TAUT;

:: PROCAL_1:th 37
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => (b2 '&' 'not' b2)) => 'not' b1 in TAUT;

:: PROCAL_1:th 38
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 'or' b2) '&' (b1 'or' b3)) => (b1 'or' (b2 '&' b3)) in TAUT;

:: PROCAL_1:th 39
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 '&' (b2 'or' b3)) => ((b1 '&' b2) 'or' (b1 '&' b3)) in TAUT;

:: PROCAL_1:th 40
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 'or' b2) '&' (b3 'or' b2)) => ((b1 '&' b3) 'or' b2) in TAUT;

:: PROCAL_1:th 41
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 'or' b2) '&' b3) => ((b1 '&' b3) 'or' (b2 '&' b3)) in TAUT;

:: PROCAL_1:th 42
theorem
for b1, b2 being Element of CQC-WFF
      st b1 in TAUT
   holds b1 'or' b2 in TAUT;

:: PROCAL_1:th 43
theorem
for b1, b2 being Element of CQC-WFF
      st b1 in TAUT
   holds b2 'or' b1 in TAUT;

:: PROCAL_1:th 44
theorem
for b1, b2 being Element of CQC-WFF
      st b1 '&' b2 in TAUT
   holds b1 in TAUT;

:: PROCAL_1:th 45
theorem
for b1, b2 being Element of CQC-WFF
      st b1 '&' b2 in TAUT
   holds b2 in TAUT;

:: PROCAL_1:th 46
theorem
for b1, b2 being Element of CQC-WFF
      st b1 '&' b2 in TAUT
   holds b1 'or' b2 in TAUT;

:: PROCAL_1:th 47
theorem
for b1, b2 being Element of CQC-WFF
      st b1 in TAUT & b2 in TAUT
   holds b1 '&' b2 in TAUT;

:: PROCAL_1:th 48
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 => b2 in TAUT
   holds (b1 'or' b3) => (b2 'or' b3) in TAUT;

:: PROCAL_1:th 49
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 => b2 in TAUT
   holds (b3 'or' b1) => (b3 'or' b2) in TAUT;

:: PROCAL_1:th 50
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 => b2 in TAUT
   holds (b3 '&' b1) => (b3 '&' b2) in TAUT;

:: PROCAL_1:th 51
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 => b2 in TAUT
   holds (b1 '&' b3) => (b2 '&' b3) in TAUT;

:: PROCAL_1:th 52
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 => b2 in TAUT & b1 => b3 in TAUT
   holds b1 => (b2 '&' b3) in TAUT;

:: PROCAL_1:th 53
theorem
for b1, b2, b3 being Element of CQC-WFF
      st b1 => b2 in TAUT & b3 => b2 in TAUT
   holds (b1 'or' b3) => b2 in TAUT;

:: PROCAL_1:th 54
theorem
for b1, b2 being Element of CQC-WFF
      st b1 'or' b2 in TAUT & 'not' b1 in TAUT
   holds b2 in TAUT;

:: PROCAL_1:th 55
theorem
for b1, b2 being Element of CQC-WFF
      st b1 'or' b2 in TAUT & 'not' b2 in TAUT
   holds b1 in TAUT;

:: PROCAL_1:th 56
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 => b2 in TAUT & b3 => b4 in TAUT
   holds (b1 '&' b3) => (b2 '&' b4) in TAUT;

:: PROCAL_1:th 57
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
      st b1 => b2 in TAUT & b3 => b4 in TAUT
   holds (b1 'or' b3) => (b2 'or' b4) in TAUT;

:: PROCAL_1:th 58
theorem
for b1, b2 being Element of CQC-WFF
      st (b1 '&' 'not' b2) => 'not' b1 in TAUT
   holds b1 => b2 in TAUT;