:: Calculus of Propositions
:: by Jan Popio{\l}ek and Andrzej Trybulec
::
:: Received October 23, 1990
:: Copyright (c) 1990 Association of Mizar Users
theorem Th1: :: PROCAL_1:1
Lm1:
for p, q being Element of CQC-WFF holds p 'or' q = ('not' p) => q
theorem Th2: :: PROCAL_1:2
theorem Th3: :: PROCAL_1:3
theorem Th4: :: PROCAL_1:4
theorem Th5: :: PROCAL_1:5
theorem Th6: :: PROCAL_1:6
theorem Th7: :: PROCAL_1:7
theorem Th8: :: PROCAL_1:8
theorem :: PROCAL_1:9
theorem :: PROCAL_1:10
theorem Th11: :: PROCAL_1:11
theorem :: PROCAL_1:12
theorem :: PROCAL_1:13
theorem :: PROCAL_1:14
Lm2:
for p, q being Element of CQC-WFF holds (p '&' q) => (('not' ('not' p)) '&' q) in TAUT
Lm3:
for p, q being Element of CQC-WFF holds (('not' ('not' p)) '&' q) => (p '&' q) in TAUT
theorem Th15: :: PROCAL_1:15
theorem Th16: :: PROCAL_1:16
theorem Th17: :: PROCAL_1:17
theorem Th18: :: PROCAL_1:18
theorem Th19: :: PROCAL_1:19
theorem Th20: :: PROCAL_1:20
theorem Th21: :: PROCAL_1:21
theorem :: PROCAL_1:22
theorem :: PROCAL_1:23
theorem :: PROCAL_1:24
theorem Th25: :: PROCAL_1:25
theorem :: PROCAL_1:26
theorem Th27: :: PROCAL_1:27
theorem Th28: :: PROCAL_1:28
theorem :: PROCAL_1:29
Lm4:
for p, q being Element of CQC-WFF st p in TAUT & q in TAUT holds
p '&' q in TAUT
theorem :: PROCAL_1:30
theorem :: PROCAL_1:31
theorem Th32: :: PROCAL_1:32
theorem Th33: :: PROCAL_1:33
theorem :: PROCAL_1:34
theorem Th35: :: PROCAL_1:35
theorem Th36: :: PROCAL_1:36
theorem :: PROCAL_1:37
theorem :: PROCAL_1:38
theorem :: PROCAL_1:39
theorem Th40: :: PROCAL_1:40
Lm5:
for p, q, r being Element of CQC-WFF st p => q in TAUT holds
(r '&' p) => (r '&' q) in TAUT
Lm6:
for p, q, r being Element of CQC-WFF st p => q in TAUT holds
(p 'or' r) => (q 'or' r) in TAUT
Lm7:
for p, q, r being Element of CQC-WFF st p => q in TAUT holds
(r 'or' p) => (r 'or' q) in TAUT
theorem :: PROCAL_1:41
theorem :: PROCAL_1:42
theorem :: PROCAL_1:43
theorem :: PROCAL_1:44
theorem :: PROCAL_1:45
theorem :: PROCAL_1:46
theorem :: PROCAL_1:47
theorem :: PROCAL_1:48
theorem :: PROCAL_1:49
theorem :: PROCAL_1:50
theorem Th51: :: PROCAL_1:51
theorem :: PROCAL_1:52
theorem :: PROCAL_1:53
theorem :: PROCAL_1:54
theorem :: PROCAL_1:55
theorem :: PROCAL_1:56
theorem :: PROCAL_1:57
theorem :: PROCAL_1:58