:: INTPRO_1 semantic presentation
:: deftheorem Def1 defines with_FALSUM INTPRO_1:def 1 :
:: deftheorem Def2 defines with_int_implication INTPRO_1:def 2 :
:: deftheorem Def3 defines with_int_conjunction INTPRO_1:def 3 :
:: deftheorem Def4 defines with_int_disjunction INTPRO_1:def 4 :
:: deftheorem Def5 defines with_int_propositional_variables INTPRO_1:def 5 :
:: deftheorem Def6 defines with_modal_operator INTPRO_1:def 6 :
:: deftheorem Def7 defines MC-closed INTPRO_1:def 7 :
Lemma22:
for E being set st E is MC-closed holds
not E is empty
:: deftheorem Def8 defines MC-wff INTPRO_1:def 8 :
:: deftheorem Def9 defines FALSUM INTPRO_1:def 9 :
:: deftheorem Def10 defines => INTPRO_1:def 10 :
:: deftheorem Def11 defines '&' INTPRO_1:def 11 :
:: deftheorem Def12 defines 'or' INTPRO_1:def 12 :
:: deftheorem Def13 defines Nes INTPRO_1:def 13 :
:: deftheorem Def14 defines IPC_theory INTPRO_1:def 14 :
:: deftheorem Def15 defines CnIPC INTPRO_1:def 15 :
:: deftheorem Def16 defines IPC-Taut INTPRO_1:def 16 :
:: deftheorem Def17 defines neg INTPRO_1:def 17 :
:: deftheorem Def18 defines IVERUM INTPRO_1:def 18 :
theorem Th1: :: INTPRO_1:1
theorem Th2: :: INTPRO_1:2
theorem Th3: :: INTPRO_1:3
theorem Th4: :: INTPRO_1:4
theorem Th5: :: INTPRO_1:5
theorem Th6: :: INTPRO_1:6
theorem Th7: :: INTPRO_1:7
theorem Th8: :: INTPRO_1:8
theorem Th9: :: INTPRO_1:9
theorem Th10: :: INTPRO_1:10
theorem Th11: :: INTPRO_1:11
theorem Th12: :: INTPRO_1:12
theorem Th13: :: INTPRO_1:13
Lemma84:
for X being Subset of MC-wff holds CnIPC (CnIPC X) c= CnIPC X
theorem Th14: :: INTPRO_1:14
Lemma85:
for X being Subset of MC-wff holds CnIPC X is IPC_theory
theorem Th15: :: INTPRO_1:15
theorem Th16: :: INTPRO_1:16
theorem Th17: :: INTPRO_1:17
theorem Th18: :: INTPRO_1:18
theorem Th19: :: INTPRO_1:19
theorem Th20: :: INTPRO_1:20
theorem Th21: :: INTPRO_1:21
theorem Th22: :: INTPRO_1:22
theorem Th23: :: INTPRO_1:23
theorem Th24: :: INTPRO_1:24
theorem Th25: :: INTPRO_1:25
theorem Th26: :: INTPRO_1:26
Lemma94:
for q, r, p, s being Element of MC-wff holds (((q => r) => (p => r)) => s) => ((p => q) => s) in IPC-Taut
theorem Th27: :: INTPRO_1:27
theorem Th28: :: INTPRO_1:28
theorem Th29: :: INTPRO_1:29
theorem Th30: :: INTPRO_1:30
theorem Th31: :: INTPRO_1:31
theorem Th32: :: INTPRO_1:32
theorem Th33: :: INTPRO_1:33
theorem Th34: :: INTPRO_1:34
theorem Th35: :: INTPRO_1:35
theorem Th36: :: INTPRO_1:36
theorem Th37: :: INTPRO_1:37
theorem Th38: :: INTPRO_1:38
theorem Th39: :: INTPRO_1:39
theorem Th40: :: INTPRO_1:40
theorem Th41: :: INTPRO_1:41
theorem Th42: :: INTPRO_1:42
theorem Th43: :: INTPRO_1:43
theorem Th44: :: INTPRO_1:44
theorem Th45: :: INTPRO_1:45
theorem Th46: :: INTPRO_1:46
theorem Th47: :: INTPRO_1:47
theorem Th48: :: INTPRO_1:48
theorem Th49: :: INTPRO_1:49
theorem Th50: :: INTPRO_1:50
theorem Th51: :: INTPRO_1:51
Lemma117:
for p, q, s being Element of MC-wff holds ((p '&' q) '&' s) => q in IPC-Taut
Lemma118:
for p, q, s being Element of MC-wff holds (((p '&' q) '&' s) '&' ((p '&' q) '&' s)) => (((p '&' q) '&' s) '&' q) in IPC-Taut
Lemma119:
for p, q, s being Element of MC-wff holds ((p '&' q) '&' s) => (((p '&' q) '&' s) '&' q) in IPC-Taut
Lemma120:
for p, q, s being Element of MC-wff holds ((p '&' q) '&' s) => (p '&' s) in IPC-Taut
Lemma121:
for p, q, s being Element of MC-wff holds (((p '&' q) '&' s) '&' q) => ((p '&' s) '&' q) in IPC-Taut
Lemma122:
for p, q, s being Element of MC-wff holds ((p '&' q) '&' s) => ((p '&' s) '&' q) in IPC-Taut
Lemma123:
for p, s, q being Element of MC-wff holds ((p '&' s) '&' q) => ((s '&' p) '&' q) in IPC-Taut
Lemma124:
for p, q, s being Element of MC-wff holds ((p '&' q) '&' s) => ((s '&' p) '&' q) in IPC-Taut
Lemma125:
for p, q, s being Element of MC-wff holds ((p '&' q) '&' s) => ((s '&' q) '&' p) in IPC-Taut
Lemma126:
for p, q, s being Element of MC-wff holds ((p '&' q) '&' s) => (p '&' (s '&' q)) in IPC-Taut
Lemma127:
for p, s, q being Element of MC-wff holds (p '&' (s '&' q)) => (p '&' (q '&' s)) in IPC-Taut
theorem Th52: :: INTPRO_1:52
Lemma128:
for p, q, s being Element of MC-wff holds (p '&' (q '&' s)) => ((s '&' q) '&' p) in IPC-Taut
Lemma129:
for s, q, p being Element of MC-wff holds ((s '&' q) '&' p) => ((q '&' s) '&' p) in IPC-Taut
Lemma130:
for p, q, s being Element of MC-wff holds (p '&' (q '&' s)) => ((q '&' s) '&' p) in IPC-Taut
Lemma131:
for p, q, s being Element of MC-wff holds (p '&' (q '&' s)) => ((p '&' s) '&' q) in IPC-Taut
Lemma132:
for p, q, s being Element of MC-wff holds (p '&' (q '&' s)) => (p '&' (s '&' q)) in IPC-Taut
theorem Th53: :: INTPRO_1:53
theorem Th54: :: INTPRO_1:54
theorem Th55: :: INTPRO_1:55
theorem Th56: :: INTPRO_1:56
theorem Th57: :: INTPRO_1:57
theorem Th58: :: INTPRO_1:58
theorem Th59: :: INTPRO_1:59
theorem Th60: :: INTPRO_1:60
theorem Th61: :: INTPRO_1:61
theorem Th62: :: INTPRO_1:62
theorem Th63: :: INTPRO_1:63
theorem Th64: :: INTPRO_1:64
theorem Th65: :: INTPRO_1:65
theorem Th66: :: INTPRO_1:66
:: deftheorem Def19 defines CPC_theory INTPRO_1:def 19 :
theorem Th67: :: INTPRO_1:67
:: deftheorem Def20 defines CnCPC INTPRO_1:def 20 :
:: deftheorem Def21 defines CPC-Taut INTPRO_1:def 21 :
theorem Th68: :: INTPRO_1:68
theorem Th69: :: INTPRO_1:69
theorem Th70: :: INTPRO_1:70
theorem Th71: :: INTPRO_1:71
theorem Th72: :: INTPRO_1:72
theorem Th73: :: INTPRO_1:73
Lemma151:
for X being Subset of MC-wff holds CnCPC (CnCPC X) c= CnCPC X
theorem Th74: :: INTPRO_1:74
Lemma152:
for X being Subset of MC-wff holds CnCPC X is CPC_theory
theorem Th75: :: INTPRO_1:75
theorem Th76: :: INTPRO_1:76
theorem Th77: :: INTPRO_1:77
:: deftheorem Def22 defines S4_theory INTPRO_1:def 22 :
theorem Th78: :: INTPRO_1:78
theorem Th79: :: INTPRO_1:79
:: deftheorem Def23 defines CnS4 INTPRO_1:def 23 :
:: deftheorem Def24 defines S4-Taut INTPRO_1:def 24 :
theorem Th80: :: INTPRO_1:80
theorem Th81: :: INTPRO_1:81
theorem Th82: :: INTPRO_1:82
theorem Th83: :: INTPRO_1:83
theorem Th84: :: INTPRO_1:84
theorem Th85: :: INTPRO_1:85
theorem Th86: :: INTPRO_1:86
theorem Th87: :: INTPRO_1:87
theorem Th88: :: INTPRO_1:88
theorem Th89: :: INTPRO_1:89
theorem Th90: :: INTPRO_1:90
Lemma168:
for X being Subset of MC-wff holds CnS4 (CnS4 X) c= CnS4 X
theorem Th91: :: INTPRO_1:91
Lemma169:
for X being Subset of MC-wff holds CnS4 X is S4_theory
theorem Th92: :: INTPRO_1:92
theorem Th93: :: INTPRO_1:93
theorem Th94: :: INTPRO_1:94
theorem Th95: :: INTPRO_1:95