:: LUKASI_1 semantic presentation

theorem Th1: :: LUKASI_1:1
for p, q, r being Element of CQC-WFF holds (p => q) => ((q => r) => (p => r)) in TAUT
proof end;

theorem Th2: :: LUKASI_1:2
for p, q, r being Element of CQC-WFF st p => q in TAUT holds
(q => r) => (p => r) in TAUT
proof end;

theorem Th3: :: LUKASI_1:3
for p, q, r being Element of CQC-WFF st p => q in TAUT & q => r in TAUT holds
p => r in TAUT
proof end;

theorem Th4: :: LUKASI_1:4
for p being Element of CQC-WFF holds p => p in TAUT
proof end;

Lemma20: for q, r, p, s being Element of CQC-WFF holds (((q => r) => (p => r)) => s) => ((p => q) => s) in TAUT
proof end;

Lemma21: for p, q, r, s being Element of CQC-WFF holds (p => (q => r)) => ((s => q) => (p => (s => r))) in TAUT
proof end;

Lemma22: for p, q, r, s being Element of CQC-WFF holds (p => q) => (((p => r) => s) => ((q => r) => s)) in TAUT
proof end;

Lemma23: for t, p, r, s, q being Element of CQC-WFF holds (t => ((p => r) => s)) => ((p => q) => (t => ((q => r) => s))) in TAUT
proof end;

Lemma24: for p, q, r being Element of CQC-WFF holds ((('not' p) => q) => r) => (p => r) in TAUT
proof end;

Lemma25: for p, r, s, q being Element of CQC-WFF holds p => (((('not' p) => r) => s) => ((q => r) => s)) in TAUT
proof end;

Lemma26: for q, p being Element of CQC-WFF holds (q => ((('not' p) => p) => p)) => ((('not' p) => p) => p) in TAUT
proof end;

Lemma27: for t, p being Element of CQC-WFF holds t => ((('not' p) => p) => p) in TAUT
proof end;

Lemma28: for p, q, t being Element of CQC-WFF holds (('not' p) => q) => (t => ((q => p) => p)) in TAUT
proof end;

Lemma29: for t, q, p, r being Element of CQC-WFF holds ((t => ((q => p) => p)) => r) => ((('not' p) => q) => r) in TAUT
proof end;

Lemma30: for p, q being Element of CQC-WFF holds (('not' p) => q) => ((q => p) => p) in TAUT
proof end;

Lemma31: for p, q being Element of CQC-WFF holds p => ((q => p) => p) in TAUT
proof end;

theorem Th5: :: LUKASI_1:5
for q, p being Element of CQC-WFF holds q => (p => q) in TAUT
proof end;

theorem Th6: :: LUKASI_1:6
for p, q, r being Element of CQC-WFF holds ((p => q) => r) => (q => r) in TAUT
proof end;

theorem Th7: :: LUKASI_1:7
for q, p being Element of CQC-WFF holds q => ((q => p) => p) in TAUT
proof end;

theorem Th8: :: LUKASI_1:8
for s, q, p being Element of CQC-WFF holds (s => (q => p)) => (q => (s => p)) in TAUT
proof end;

theorem Th9: :: LUKASI_1:9
for q, r, p being Element of CQC-WFF holds (q => r) => ((p => q) => (p => r)) in TAUT
proof end;

Lemma37: for q, s, p, r being Element of CQC-WFF holds ((q => (s => p)) => r) => ((s => (q => p)) => r) in TAUT
proof end;

Lemma38: for p, q being Element of CQC-WFF holds ((p => q) => p) => p in TAUT
proof end;

Lemma39: for p, r, s, q being Element of CQC-WFF holds ((p => r) => s) => ((p => q) => ((q => r) => s)) in TAUT
proof end;

Lemma40: for p, q, r being Element of CQC-WFF holds ((p => q) => r) => ((r => p) => p) in TAUT
proof end;

Lemma41: for r, p, s, q being Element of CQC-WFF holds (((r => p) => p) => s) => (((p => q) => r) => s) in TAUT
proof end;

Lemma42: for q, r, p being Element of CQC-WFF holds ((q => r) => p) => ((q => p) => p) in TAUT
proof end;

theorem Th10: :: LUKASI_1:10
for q, r being Element of CQC-WFF holds (q => (q => r)) => (q => r) in TAUT
proof end;

Lemma44: for q, s, r, p being Element of CQC-WFF holds (q => s) => (((q => r) => p) => ((s => p) => p)) in TAUT
proof end;

Lemma45: for q, r, p, s being Element of CQC-WFF holds ((q => r) => p) => ((q => s) => ((s => p) => p)) in TAUT
proof end;

Lemma46: for q, s, p, r being Element of CQC-WFF holds (q => s) => ((s => (p => (q => r))) => (p => (q => r))) in TAUT
proof end;

Lemma47: for s, p, q, r being Element of CQC-WFF holds (s => (p => (q => r))) => ((q => s) => (p => (q => r))) in TAUT
proof end;

theorem Th11: :: LUKASI_1:11
for p, q, r being Element of CQC-WFF holds (p => (q => r)) => ((p => q) => (p => r)) in TAUT
proof end;

theorem Th12: :: LUKASI_1:12
for p being Element of CQC-WFF holds ('not' VERUM ) => p in TAUT
proof end;

theorem Th13: :: LUKASI_1:13
for q, p being Element of CQC-WFF st q in TAUT holds
p => q in TAUT
proof end;

theorem Th14: :: LUKASI_1:14
for p, q being Element of CQC-WFF st p in TAUT holds
(p => q) => q in TAUT
proof end;

theorem Th15: :: LUKASI_1:15
for s, q, p being Element of CQC-WFF st s => (q => p) in TAUT holds
q => (s => p) in TAUT
proof end;

theorem Th16: :: LUKASI_1:16
for s, q, p being Element of CQC-WFF st s => (q => p) in TAUT & q in TAUT holds
s => p in TAUT
proof end;

theorem Th17: :: LUKASI_1:17
for s, q, p being Element of CQC-WFF st s => (q => p) in TAUT & q in TAUT & s in TAUT holds
p in TAUT
proof end;

theorem Th18: :: LUKASI_1:18
for q, r being Element of CQC-WFF st q => (q => r) in TAUT holds
q => r in TAUT
proof end;

theorem Th19: :: LUKASI_1:19
for p, q, r being Element of CQC-WFF st p => (q => r) in TAUT holds
(p => q) => (p => r) in TAUT
proof end;

theorem Th20: :: LUKASI_1:20
for p, q, r being Element of CQC-WFF st p => (q => r) in TAUT & p => q in TAUT holds
p => r in TAUT
proof end;

theorem Th21: :: LUKASI_1:21
for p, q, r being Element of CQC-WFF st p => (q => r) in TAUT & p => q in TAUT & p in TAUT holds
r in TAUT
proof end;

theorem Th22: :: LUKASI_1:22
for p, q, r, s being Element of CQC-WFF st p => (q => r) in TAUT & p => (r => s) in TAUT holds
p => (q => s) in TAUT
proof end;

theorem Th23: :: LUKASI_1:23
for p being Element of CQC-WFF holds p => VERUM in TAUT by Th8, CQC_THE1:77;

Lemma56: for p being Element of CQC-WFF holds ('not' p) => (p => ('not' VERUM )) in TAUT
proof end;

Lemma57: for p being Element of CQC-WFF holds (('not' p) => ('not' VERUM )) => p in TAUT
proof end;

theorem Th24: :: LUKASI_1:24
for p, q being Element of CQC-WFF holds (('not' p) => ('not' q)) => (q => p) in TAUT
proof end;

theorem Th25: :: LUKASI_1:25
for p being Element of CQC-WFF holds ('not' ('not' p)) => p in TAUT
proof end;

E60: now
let p be Element of CQC-WFF ;
('not' ('not' p)) => p in TAUT by Lemma44;
then E16: (p => ('not' VERUM )) => (('not' ('not' p)) => ('not' VERUM )) in TAUT by ;
(('not' ('not' p)) => ('not' VERUM )) => ('not' p) in TAUT by Lemma42;
hence (p => ('not' VERUM )) => ('not' p) in TAUT by Lemma46, ;
end;

theorem Th26: :: LUKASI_1:26
for p, q being Element of CQC-WFF holds (p => q) => (('not' q) => ('not' p)) in TAUT
proof end;

theorem Th27: :: LUKASI_1:27
for p being Element of CQC-WFF holds p => ('not' ('not' p)) in TAUT
proof end;

theorem Th28: :: LUKASI_1:28
for p, q being Element of CQC-WFF holds
( (('not' ('not' p)) => q) => (p => q) in TAUT & (p => q) => (('not' ('not' p)) => q) in TAUT )
proof end;

theorem Th29: :: LUKASI_1:29
for p, q being Element of CQC-WFF holds
( (p => ('not' ('not' q))) => (p => q) in TAUT & (p => q) => (p => ('not' ('not' q))) in TAUT )
proof end;

theorem Th30: :: LUKASI_1:30
for p, q being Element of CQC-WFF holds (p => ('not' q)) => (q => ('not' p)) in TAUT
proof end;

theorem Th31: :: LUKASI_1:31
for p, q being Element of CQC-WFF holds (('not' p) => q) => (('not' q) => p) in TAUT
proof end;

theorem Th32: :: LUKASI_1:32
for p being Element of CQC-WFF holds (p => ('not' p)) => ('not' p) in TAUT
proof end;

theorem Th33: :: LUKASI_1:33
for p, q being Element of CQC-WFF holds ('not' p) => (p => q) in TAUT
proof end;

theorem Th34: :: LUKASI_1:34
for p, q being Element of CQC-WFF holds
( p => q in TAUT iff ('not' q) => ('not' p) in TAUT )
proof end;

theorem Th35: :: LUKASI_1:35
for p, q being Element of CQC-WFF st ('not' p) => ('not' q) in TAUT holds
q => p in TAUT by Th16;

theorem Th36: :: LUKASI_1:36
for p being Element of CQC-WFF holds
( p in TAUT iff 'not' ('not' p) in TAUT )
proof end;

theorem Th37: :: LUKASI_1:37
for p, q being Element of CQC-WFF holds
( p => q in TAUT iff p => ('not' ('not' q)) in TAUT )
proof end;

theorem Th38: :: LUKASI_1:38
for p, q being Element of CQC-WFF holds
( p => q in TAUT iff ('not' ('not' p)) => q in TAUT )
proof end;

theorem Th39: :: LUKASI_1:39
for p, q being Element of CQC-WFF st p => ('not' q) in TAUT holds
q => ('not' p) in TAUT
proof end;

theorem Th40: :: LUKASI_1:40
for p, q being Element of CQC-WFF st ('not' p) => q in TAUT holds
('not' q) => p in TAUT
proof end;

theorem Th41: :: LUKASI_1:41
for p, q, r being Element of CQC-WFF holds |- (p => q) => ((q => r) => (p => r))
proof end;

theorem Th42: :: LUKASI_1:42
for p, q, r being Element of CQC-WFF st |- p => q holds
|- (q => r) => (p => r)
proof end;

theorem Th43: :: LUKASI_1:43
for p, q, r being Element of CQC-WFF st |- p => q & |- q => r holds
|- p => r
proof end;

theorem Th44: :: LUKASI_1:44
for p being Element of CQC-WFF holds |- p => p
proof end;

theorem Th45: :: LUKASI_1:45
for p, q being Element of CQC-WFF holds |- p => (q => p)
proof end;

theorem Th46: :: LUKASI_1:46
for p, q being Element of CQC-WFF st |- p holds
|- q => p
proof end;

theorem Th47: :: LUKASI_1:47
for s, q, p being Element of CQC-WFF holds |- (s => (q => p)) => (q => (s => p))
proof end;

theorem Th48: :: LUKASI_1:48
for p, q, r being Element of CQC-WFF st |- p => (q => r) holds
|- q => (p => r)
proof end;

theorem Th49: :: LUKASI_1:49
for p, q, r being Element of CQC-WFF st |- p => (q => r) & |- q holds
|- p => r
proof end;

theorem Th50: :: LUKASI_1:50
for p being Element of CQC-WFF holds
( |- p => VERUM & |- ('not' VERUM ) => p )
proof end;

theorem Th51: :: LUKASI_1:51
for p, q being Element of CQC-WFF holds |- p => ((p => q) => q)
proof end;

theorem Th52: :: LUKASI_1:52
for q, r being Element of CQC-WFF holds |- (q => (q => r)) => (q => r)
proof end;

theorem Th53: :: LUKASI_1:53
for q, r being Element of CQC-WFF st |- q => (q => r) holds
|- q => r
proof end;

theorem Th54: :: LUKASI_1:54
for p, q, r being Element of CQC-WFF holds |- (p => (q => r)) => ((p => q) => (p => r))
proof end;

theorem Th55: :: LUKASI_1:55
for p, q, r being Element of CQC-WFF st |- p => (q => r) holds
|- (p => q) => (p => r)
proof end;

theorem Th56: :: LUKASI_1:56
for p, q, r being Element of CQC-WFF st |- p => (q => r) & |- p => q holds
|- p => r
proof end;

theorem Th57: :: LUKASI_1:57
for p, q, r being Element of CQC-WFF holds |- ((p => q) => r) => (q => r)
proof end;

theorem Th58: :: LUKASI_1:58
for p, q, r being Element of CQC-WFF st |- (p => q) => r holds
|- q => r
proof end;

theorem Th59: :: LUKASI_1:59
for p, q, r being Element of CQC-WFF holds |- (p => q) => ((r => p) => (r => q))
proof end;

theorem Th60: :: LUKASI_1:60
for p, q, r being Element of CQC-WFF st |- p => q holds
|- (r => p) => (r => q)
proof end;

theorem Th61: :: LUKASI_1:61
for p, q being Element of CQC-WFF holds |- (p => q) => (('not' q) => ('not' p))
proof end;

theorem Th62: :: LUKASI_1:62
for p, q being Element of CQC-WFF holds |- (('not' p) => ('not' q)) => (q => p)
proof end;

theorem Th63: :: LUKASI_1:63
for p, q being Element of CQC-WFF holds
( |- ('not' p) => ('not' q) iff |- q => p )
proof end;

theorem Th64: :: LUKASI_1:64
for p being Element of CQC-WFF holds |- p => ('not' ('not' p))
proof end;

theorem Th65: :: LUKASI_1:65
for p being Element of CQC-WFF holds |- ('not' ('not' p)) => p
proof end;

theorem Th66: :: LUKASI_1:66
for p being Element of CQC-WFF holds
( |- 'not' ('not' p) iff |- p )
proof end;

theorem Th67: :: LUKASI_1:67
for p, q being Element of CQC-WFF holds |- (('not' ('not' p)) => q) => (p => q)
proof end;

theorem Th68: :: LUKASI_1:68
for p, q being Element of CQC-WFF holds
( |- ('not' ('not' p)) => q iff |- p => q )
proof end;

theorem Th69: :: LUKASI_1:69
for p, q being Element of CQC-WFF holds |- (p => ('not' ('not' q))) => (p => q)
proof end;

theorem Th70: :: LUKASI_1:70
for p, q being Element of CQC-WFF holds
( |- p => ('not' ('not' q)) iff |- p => q )
proof end;

theorem Th71: :: LUKASI_1:71
for p, q being Element of CQC-WFF holds |- (p => ('not' q)) => (q => ('not' p))
proof end;

theorem Th72: :: LUKASI_1:72
for p, q being Element of CQC-WFF st |- p => ('not' q) holds
|- q => ('not' p)
proof end;

theorem Th73: :: LUKASI_1:73
for p, q being Element of CQC-WFF holds |- (('not' p) => q) => (('not' q) => p)
proof end;

theorem Th74: :: LUKASI_1:74
for p, q being Element of CQC-WFF st |- ('not' p) => q holds
|- ('not' q) => p
proof end;

theorem Th75: :: LUKASI_1:75
for p, q, r being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => q holds
X |- (q => r) => (p => r)
proof end;

theorem Th76: :: LUKASI_1:76
for p, q, r being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => q & X |- q => r holds
X |- p => r
proof end;

theorem Th77: :: LUKASI_1:77
for p being Element of CQC-WFF
for X being Subset of CQC-WFF holds X |- p => p
proof end;

theorem Th78: :: LUKASI_1:78
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p holds
X |- q => p
proof end;

theorem Th79: :: LUKASI_1:79
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p holds
X |- (p => q) => q
proof end;

theorem Th80: :: LUKASI_1:80
for p, q, r being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => (q => r) holds
X |- q => (p => r)
proof end;

theorem Th81: :: LUKASI_1:81
for p, q, r being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => (q => r) & X |- q holds
X |- p => r
proof end;

theorem Th82: :: LUKASI_1:82
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => (p => q) holds
X |- p => q
proof end;

theorem Th83: :: LUKASI_1:83
for p, q, r being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- (p => q) => r holds
X |- q => r
proof end;

theorem Th84: :: LUKASI_1:84
for p, q, r being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => (q => r) holds
X |- (p => q) => (p => r)
proof end;

theorem Th85: :: LUKASI_1:85
for p, q, r being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => (q => r) & X |- p => q holds
X |- p => r
proof end;

theorem Th86: :: LUKASI_1:86
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF holds
( X |- ('not' p) => ('not' q) iff X |- q => p )
proof end;

theorem Th87: :: LUKASI_1:87
for p being Element of CQC-WFF
for X being Subset of CQC-WFF holds
( X |- 'not' ('not' p) iff X |- p )
proof end;

theorem Th88: :: LUKASI_1:88
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF holds
( X |- p => ('not' ('not' q)) iff X |- p => q )
proof end;

theorem Th89: :: LUKASI_1:89
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF holds
( X |- ('not' ('not' p)) => q iff X |- p => q )
proof end;

theorem Th90: :: LUKASI_1:90
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => ('not' q) holds
X |- q => ('not' p)
proof end;

theorem Th91: :: LUKASI_1:91
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- ('not' p) => q holds
X |- ('not' q) => p
proof end;

theorem Th92: :: LUKASI_1:92
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- p => ('not' q) & X |- q holds
X |- 'not' p
proof end;

theorem Th93: :: LUKASI_1:93
for p, q being Element of CQC-WFF
for X being Subset of CQC-WFF st X |- ('not' p) => q & X |- 'not' q holds
X |- p
proof end;