Article LUKASI_1, MML version 4.99.1005
:: LUKASI_1:th 1
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => b2) => ((b2 => b3) => (b1 => b3)) in TAUT;
:: LUKASI_1:th 2
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => b2 in TAUT
holds (b2 => b3) => (b1 => b3) in TAUT;
:: LUKASI_1:th 3
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => b2 in TAUT & b2 => b3 in TAUT
holds b1 => b3 in TAUT;
:: LUKASI_1:th 4
theorem
for b1 being Element of CQC-WFF holds
b1 => b1 in TAUT;
:: LUKASI_1:th 5
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => (b2 => b1) in TAUT;
:: LUKASI_1:th 6
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 => b2) => b3) => (b2 => b3) in TAUT;
:: LUKASI_1:th 7
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => ((b1 => b2) => b2) in TAUT;
:: LUKASI_1:th 8
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => (b2 => b3)) => (b2 => (b1 => b3)) in TAUT;
:: LUKASI_1:th 9
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => b2) => ((b3 => b1) => (b3 => b2)) in TAUT;
:: LUKASI_1:th 10
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => (b1 => b2)) => (b1 => b2) in TAUT;
:: LUKASI_1:th 11
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => (b2 => b3)) => ((b1 => b2) => (b1 => b3)) in TAUT;
:: LUKASI_1:th 12
theorem
for b1 being Element of CQC-WFF holds
('not' VERUM) => b1 in TAUT;
:: LUKASI_1:th 13
theorem
for b1, b2 being Element of CQC-WFF
st b1 in TAUT
holds b2 => b1 in TAUT;
:: LUKASI_1:th 14
theorem
for b1, b2 being Element of CQC-WFF
st b1 in TAUT
holds (b1 => b2) => b2 in TAUT;
:: LUKASI_1:th 15
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) in TAUT
holds b2 => (b1 => b3) in TAUT;
:: LUKASI_1:th 16
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) in TAUT & b2 in TAUT
holds b1 => b3 in TAUT;
:: LUKASI_1:th 17
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) in TAUT & b2 in TAUT & b1 in TAUT
holds b3 in TAUT;
:: LUKASI_1:th 18
theorem
for b1, b2 being Element of CQC-WFF
st b1 => (b1 => b2) in TAUT
holds b1 => b2 in TAUT;
:: LUKASI_1:th 19
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) in TAUT
holds (b1 => b2) => (b1 => b3) in TAUT;
:: LUKASI_1:th 20
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) in TAUT & b1 => b2 in TAUT
holds b1 => b3 in TAUT;
:: LUKASI_1:th 21
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) in TAUT & b1 => b2 in TAUT & b1 in TAUT
holds b3 in TAUT;
:: LUKASI_1:th 22
theorem
for b1, b2, b3, b4 being Element of CQC-WFF
st b1 => (b2 => b3) in TAUT & b1 => (b3 => b4) in TAUT
holds b1 => (b2 => b4) in TAUT;
:: LUKASI_1:th 23
theorem
for b1 being Element of CQC-WFF holds
b1 => VERUM in TAUT;
:: LUKASI_1:th 24
theorem
for b1, b2 being Element of CQC-WFF holds
(('not' b1) => 'not' b2) => (b2 => b1) in TAUT;
:: LUKASI_1:th 25
theorem
for b1 being Element of CQC-WFF holds
('not' 'not' b1) => b1 in TAUT;
:: LUKASI_1:th 26
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => b2) => (('not' b2) => 'not' b1) in TAUT;
:: LUKASI_1:th 27
theorem
for b1 being Element of CQC-WFF holds
b1 => 'not' 'not' b1 in TAUT;
:: LUKASI_1:th 28
theorem
for b1, b2 being Element of CQC-WFF holds
(('not' 'not' b1) => b2) => (b1 => b2) in TAUT &
(b1 => b2) => (('not' 'not' b1) => b2) in TAUT;
:: LUKASI_1:th 29
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => 'not' 'not' b2) => (b1 => b2) in TAUT &
(b1 => b2) => (b1 => 'not' 'not' b2) in TAUT;
:: LUKASI_1:th 30
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => 'not' b2) => (b2 => 'not' b1) in TAUT;
:: LUKASI_1:th 31
theorem
for b1, b2 being Element of CQC-WFF holds
(('not' b1) => b2) => (('not' b2) => b1) in TAUT;
:: LUKASI_1:th 32
theorem
for b1 being Element of CQC-WFF holds
(b1 => 'not' b1) => 'not' b1 in TAUT;
:: LUKASI_1:th 33
theorem
for b1, b2 being Element of CQC-WFF holds
('not' b1) => (b1 => b2) in TAUT;
:: LUKASI_1:th 34
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => b2 in TAUT
iff
('not' b2) => 'not' b1 in TAUT;
:: LUKASI_1:th 35
theorem
for b1, b2 being Element of CQC-WFF
st ('not' b1) => 'not' b2 in TAUT
holds b2 => b1 in TAUT;
:: LUKASI_1:th 36
theorem
for b1 being Element of CQC-WFF holds
b1 in TAUT
iff
'not' 'not' b1 in TAUT;
:: LUKASI_1:th 37
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => b2 in TAUT
iff
b1 => 'not' 'not' b2 in TAUT;
:: LUKASI_1:th 38
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => b2 in TAUT
iff
('not' 'not' b1) => b2 in TAUT;
:: LUKASI_1:th 39
theorem
for b1, b2 being Element of CQC-WFF
st b1 => 'not' b2 in TAUT
holds b2 => 'not' b1 in TAUT;
:: LUKASI_1:th 40
theorem
for b1, b2 being Element of CQC-WFF
st ('not' b1) => b2 in TAUT
holds ('not' b2) => b1 in TAUT;
:: LUKASI_1:th 41
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => b2) => ((b2 => b3) => (b1 => b3)) is valid;
:: LUKASI_1:th 42
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => b2 is valid
holds (b2 => b3) => (b1 => b3) is valid;
:: LUKASI_1:th 43
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => b2 is valid & b2 => b3 is valid
holds b1 => b3 is valid;
:: LUKASI_1:th 44
theorem
for b1 being Element of CQC-WFF holds
b1 => b1 is valid;
:: LUKASI_1:th 45
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => (b2 => b1) is valid;
:: LUKASI_1:th 46
theorem
for b1, b2 being Element of CQC-WFF
st b1 is valid
holds b2 => b1 is valid;
:: LUKASI_1:th 47
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => (b2 => b3)) => (b2 => (b1 => b3)) is valid;
:: LUKASI_1:th 48
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) is valid
holds b2 => (b1 => b3) is valid;
:: LUKASI_1:th 49
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) is valid & b2 is valid
holds b1 => b3 is valid;
:: LUKASI_1:th 50
theorem
for b1 being Element of CQC-WFF holds
b1 => VERUM is valid & ('not' VERUM) => b1 is valid;
:: LUKASI_1:th 51
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => ((b1 => b2) => b2) is valid;
:: LUKASI_1:th 52
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => (b1 => b2)) => (b1 => b2) is valid;
:: LUKASI_1:th 53
theorem
for b1, b2 being Element of CQC-WFF
st b1 => (b1 => b2) is valid
holds b1 => b2 is valid;
:: LUKASI_1:th 54
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => (b2 => b3)) => ((b1 => b2) => (b1 => b3)) is valid;
:: LUKASI_1:th 55
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) is valid
holds (b1 => b2) => (b1 => b3) is valid;
:: LUKASI_1:th 56
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => (b2 => b3) is valid & b1 => b2 is valid
holds b1 => b3 is valid;
:: LUKASI_1:th 57
theorem
for b1, b2, b3 being Element of CQC-WFF holds
((b1 => b2) => b3) => (b2 => b3) is valid;
:: LUKASI_1:th 58
theorem
for b1, b2, b3 being Element of CQC-WFF
st (b1 => b2) => b3 is valid
holds b2 => b3 is valid;
:: LUKASI_1:th 59
theorem
for b1, b2, b3 being Element of CQC-WFF holds
(b1 => b2) => ((b3 => b1) => (b3 => b2)) is valid;
:: LUKASI_1:th 60
theorem
for b1, b2, b3 being Element of CQC-WFF
st b1 => b2 is valid
holds (b3 => b1) => (b3 => b2) is valid;
:: LUKASI_1:th 61
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => b2) => (('not' b2) => 'not' b1) is valid;
:: LUKASI_1:th 62
theorem
for b1, b2 being Element of CQC-WFF holds
(('not' b1) => 'not' b2) => (b2 => b1) is valid;
:: LUKASI_1:th 63
theorem
for b1, b2 being Element of CQC-WFF holds
('not' b1) => 'not' b2 is valid
iff
b2 => b1 is valid;
:: LUKASI_1:th 64
theorem
for b1 being Element of CQC-WFF holds
b1 => 'not' 'not' b1 is valid;
:: LUKASI_1:th 65
theorem
for b1 being Element of CQC-WFF holds
('not' 'not' b1) => b1 is valid;
:: LUKASI_1:th 66
theorem
for b1 being Element of CQC-WFF holds
'not' 'not' b1 is valid
iff
b1 is valid;
:: LUKASI_1:th 67
theorem
for b1, b2 being Element of CQC-WFF holds
(('not' 'not' b1) => b2) => (b1 => b2) is valid;
:: LUKASI_1:th 68
theorem
for b1, b2 being Element of CQC-WFF holds
('not' 'not' b1) => b2 is valid
iff
b1 => b2 is valid;
:: LUKASI_1:th 69
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => 'not' 'not' b2) => (b1 => b2) is valid;
:: LUKASI_1:th 70
theorem
for b1, b2 being Element of CQC-WFF holds
b1 => 'not' 'not' b2 is valid
iff
b1 => b2 is valid;
:: LUKASI_1:th 71
theorem
for b1, b2 being Element of CQC-WFF holds
(b1 => 'not' b2) => (b2 => 'not' b1) is valid;
:: LUKASI_1:th 72
theorem
for b1, b2 being Element of CQC-WFF
st b1 => 'not' b2 is valid
holds b2 => 'not' b1 is valid;
:: LUKASI_1:th 73
theorem
for b1, b2 being Element of CQC-WFF holds
(('not' b1) => b2) => (('not' b2) => b1) is valid;
:: LUKASI_1:th 74
theorem
for b1, b2 being Element of CQC-WFF
st ('not' b1) => b2 is valid
holds ('not' b2) => b1 is valid;
:: LUKASI_1:th 75
theorem
for b1, b2, b3 being Element of CQC-WFF
for b4 being Element of bool CQC-WFF
st b4 |- b1 => b2
holds b4 |- (b2 => b3) => (b1 => b3);
:: LUKASI_1:th 76
theorem
for b1, b2, b3 being Element of CQC-WFF
for b4 being Element of bool CQC-WFF
st b4 |- b1 => b2 & b4 |- b2 => b3
holds b4 |- b1 => b3;
:: LUKASI_1:th 77
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bool CQC-WFF holds
b2 |- b1 => b1;
:: LUKASI_1:th 78
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
st b3 |- b1
holds b3 |- b2 => b1;
:: LUKASI_1:th 79
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
st b3 |- b1
holds b3 |- (b1 => b2) => b2;
:: LUKASI_1:th 80
theorem
for b1, b2, b3 being Element of CQC-WFF
for b4 being Element of bool CQC-WFF
st b4 |- b1 => (b2 => b3)
holds b4 |- b2 => (b1 => b3);
:: LUKASI_1:th 81
theorem
for b1, b2, b3 being Element of CQC-WFF
for b4 being Element of bool CQC-WFF
st b4 |- b1 => (b2 => b3) & b4 |- b2
holds b4 |- b1 => b3;
:: LUKASI_1:th 82
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
st b3 |- b1 => (b1 => b2)
holds b3 |- b1 => b2;
:: LUKASI_1:th 83
theorem
for b1, b2, b3 being Element of CQC-WFF
for b4 being Element of bool CQC-WFF
st b4 |- (b1 => b2) => b3
holds b4 |- b2 => b3;
:: LUKASI_1:th 84
theorem
for b1, b2, b3 being Element of CQC-WFF
for b4 being Element of bool CQC-WFF
st b4 |- b1 => (b2 => b3)
holds b4 |- (b1 => b2) => (b1 => b3);
:: LUKASI_1:th 85
theorem
for b1, b2, b3 being Element of CQC-WFF
for b4 being Element of bool CQC-WFF
st b4 |- b1 => (b2 => b3) & b4 |- b1 => b2
holds b4 |- b1 => b3;
:: LUKASI_1:th 86
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF holds
b3 |- ('not' b1) => 'not' b2
iff
b3 |- b2 => b1;
:: LUKASI_1:th 87
theorem
for b1 being Element of CQC-WFF
for b2 being Element of bool CQC-WFF holds
b2 |- 'not' 'not' b1
iff
b2 |- b1;
:: LUKASI_1:th 88
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF holds
b3 |- b1 => 'not' 'not' b2
iff
b3 |- b1 => b2;
:: LUKASI_1:th 89
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF holds
b3 |- ('not' 'not' b1) => b2
iff
b3 |- b1 => b2;
:: LUKASI_1:th 90
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
st b3 |- b1 => 'not' b2
holds b3 |- b2 => 'not' b1;
:: LUKASI_1:th 91
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
st b3 |- ('not' b1) => b2
holds b3 |- ('not' b2) => b1;
:: LUKASI_1:th 92
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
st b3 |- b1 => 'not' b2 & b3 |- b2
holds b3 |- 'not' b1;
:: LUKASI_1:th 93
theorem
for b1, b2 being Element of CQC-WFF
for b3 being Element of bool CQC-WFF
st b3 |- ('not' b1) => b2 & b3 |- 'not' b2
holds b3 |- b1;