:: HILBERT1 semantic presentation
:: deftheorem Def1 defines with_VERUM HILBERT1:def 1 :
:: deftheorem Def2 defines with_implication HILBERT1:def 2 :
:: deftheorem Def3 defines with_conjunction HILBERT1:def 3 :
:: deftheorem Def4 defines with_propositional_variables HILBERT1:def 4 :
:: deftheorem Def5 defines HP-closed HILBERT1:def 5 :
Lemma20:
for D being set st D is HP-closed holds
not D is empty
:: deftheorem Def6 defines HP-WFF HILBERT1:def 6 :
:: deftheorem Def7 defines VERUM HILBERT1:def 7 :
:: deftheorem Def8 defines => HILBERT1:def 8 :
:: deftheorem Def9 defines '&' HILBERT1:def 9 :
:: deftheorem Def10 defines Hilbert_theory HILBERT1:def 10 :
:: deftheorem Def11 defines CnPos HILBERT1:def 11 :
:: deftheorem Def12 defines HP_TAUT HILBERT1:def 12 :
theorem Th1: :: HILBERT1:1
theorem Th2: :: HILBERT1:2
theorem Th3: :: HILBERT1:3
theorem Th4: :: HILBERT1:4
theorem Th5: :: HILBERT1:5
theorem Th6: :: HILBERT1:6
theorem Th7: :: HILBERT1:7
theorem Th8: :: HILBERT1:8
theorem Th9: :: HILBERT1:9
theorem Th10: :: HILBERT1:10
Lemma69:
for X being Subset of HP-WFF holds CnPos (CnPos X) c= CnPos X
theorem Th11: :: HILBERT1:11
Lemma70:
for X being Subset of HP-WFF holds CnPos X is Hilbert_theory
theorem Th12: :: HILBERT1:12
theorem Th13: :: HILBERT1:13
theorem Th14: :: HILBERT1:14
theorem Th15: :: HILBERT1:15
theorem Th16: :: HILBERT1:16
theorem Th17: :: HILBERT1:17
theorem Th18: :: HILBERT1:18
theorem Th19: :: HILBERT1:19
theorem Th20: :: HILBERT1:20
theorem Th21: :: HILBERT1:21
theorem Th22: :: HILBERT1:22
theorem Th23: :: HILBERT1:23
Lemma79:
for q, r, p, s being Element of HP-WFF holds (((q => r) => (p => r)) => s) => ((p => q) => s) in HP_TAUT
theorem Th24: :: HILBERT1:24
theorem Th25: :: HILBERT1:25
theorem Th26: :: HILBERT1:26
theorem Th27: :: HILBERT1:27
theorem Th28: :: HILBERT1:28
theorem Th29: :: HILBERT1:29
theorem Th30: :: HILBERT1:30
theorem Th31: :: HILBERT1:31
theorem Th32: :: HILBERT1:32
theorem Th33: :: HILBERT1:33
theorem Th34: :: HILBERT1:34
theorem Th35: :: HILBERT1:35
theorem Th36: :: HILBERT1:36
theorem Th37: :: HILBERT1:37
theorem Th38: :: HILBERT1:38
theorem Th39: :: HILBERT1:39
theorem Th40: :: HILBERT1:40
theorem Th41: :: HILBERT1:41
theorem Th42: :: HILBERT1:42
theorem Th43: :: HILBERT1:43
theorem Th44: :: HILBERT1:44
theorem Th45: :: HILBERT1:45
theorem Th46: :: HILBERT1:46
theorem Th47: :: HILBERT1:47
theorem Th48: :: HILBERT1:48
Lemma102:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => q in HP_TAUT
Lemma103:
for p, q, s being Element of HP-WFF holds (((p '&' q) '&' s) '&' ((p '&' q) '&' s)) => (((p '&' q) '&' s) '&' q) in HP_TAUT
Lemma104:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (((p '&' q) '&' s) '&' q) in HP_TAUT
Lemma105:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (p '&' s) in HP_TAUT
Lemma106:
for p, q, s being Element of HP-WFF holds (((p '&' q) '&' s) '&' q) => ((p '&' s) '&' q) in HP_TAUT
Lemma107:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((p '&' s) '&' q) in HP_TAUT
Lemma108:
for p, s, q being Element of HP-WFF holds ((p '&' s) '&' q) => ((s '&' p) '&' q) in HP_TAUT
Lemma109:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((s '&' p) '&' q) in HP_TAUT
Lemma110:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((s '&' q) '&' p) in HP_TAUT
Lemma111:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (p '&' (s '&' q)) in HP_TAUT
Lemma112:
for p, s, q being Element of HP-WFF holds (p '&' (s '&' q)) => (p '&' (q '&' s)) in HP_TAUT
theorem Th49: :: HILBERT1:49
Lemma113:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((s '&' q) '&' p) in HP_TAUT
Lemma114:
for s, q, p being Element of HP-WFF holds ((s '&' q) '&' p) => ((q '&' s) '&' p) in HP_TAUT
Lemma115:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((q '&' s) '&' p) in HP_TAUT
Lemma116:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((p '&' s) '&' q) in HP_TAUT
Lemma117:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => (p '&' (s '&' q)) in HP_TAUT
theorem Th50: :: HILBERT1:50