:: LUKASI_1 semantic presentation begin theorem Th1: :: LUKASI_1:1 for A being QC-alphabet for p, q, r being Element of CQC-WFF A holds (p => q) => ((q => r) => (p => r)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A holds (p => q) => ((q => r) => (p => r)) in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: (p => q) => ((q => r) => (p => r)) in TAUT A (p => q) => (('not' (q '&' ('not' r))) => ('not' (p '&' ('not' r)))) in TAUT A by CQC_THE1:44; then (p => q) => ((q => r) => ('not' (p '&' ('not' r)))) in TAUT A by QC_LANG2:def_2; hence (p => q) => ((q => r) => (p => r)) in TAUT A by QC_LANG2:def_2; ::_thesis: verum end; theorem Th2: :: LUKASI_1:2 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => q in TAUT A holds (q => r) => (p => r) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => q in TAUT A holds (q => r) => (p => r) in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => q in TAUT A implies (q => r) => (p => r) in TAUT A ) assume A1: p => q in TAUT A ; ::_thesis: (q => r) => (p => r) in TAUT A (p => q) => ((q => r) => (p => r)) in TAUT A by Th1; hence (q => r) => (p => r) in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; theorem Th3: :: LUKASI_1:3 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => q in TAUT A & q => r in TAUT A holds p => r in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => q in TAUT A & q => r in TAUT A holds p => r in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => q in TAUT A & q => r in TAUT A implies p => r in TAUT A ) assume that A1: p => q in TAUT A and A2: q => r in TAUT A ; ::_thesis: p => r in TAUT A (p => q) => ((q => r) => (p => r)) in TAUT A by Th1; then (q => r) => (p => r) in TAUT A by A1, CQC_THE1:46; hence p => r in TAUT A by A2, CQC_THE1:46; ::_thesis: verum end; theorem Th4: :: LUKASI_1:4 for A being QC-alphabet for p being Element of CQC-WFF A holds p => p in TAUT A proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds p => p in TAUT A let p be Element of CQC-WFF A; ::_thesis: p => p in TAUT A ( (('not' p) => p) => p in TAUT A & p => (('not' p) => p) in TAUT A ) by CQC_THE1:42, CQC_THE1:43; hence p => p in TAUT A by Th3; ::_thesis: verum end; Lm1: for A being QC-alphabet for q, r, p, s being Element of CQC-WFF A holds (((q => r) => (p => r)) => s) => ((p => q) => s) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, r, p, s being Element of CQC-WFF A holds (((q => r) => (p => r)) => s) => ((p => q) => s) in TAUT A let q, r, p, s be Element of CQC-WFF A; ::_thesis: (((q => r) => (p => r)) => s) => ((p => q) => s) in TAUT A (p => q) => ((q => r) => (p => r)) in TAUT A by Th1; hence (((q => r) => (p => r)) => s) => ((p => q) => s) in TAUT A by Th2; ::_thesis: verum end; Lm2: for A being QC-alphabet for p, q, r, s being Element of CQC-WFF A holds (p => (q => r)) => ((s => q) => (p => (s => r))) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r, s being Element of CQC-WFF A holds (p => (q => r)) => ((s => q) => (p => (s => r))) in TAUT A let p, q, r, s be Element of CQC-WFF A; ::_thesis: (p => (q => r)) => ((s => q) => (p => (s => r))) in TAUT A ( ((((q => r) => (s => r)) => (p => (s => r))) => ((s => q) => (p => (s => r)))) => ((p => (q => r)) => ((s => q) => (p => (s => r)))) in TAUT A & (((q => r) => (s => r)) => (p => (s => r))) => ((s => q) => (p => (s => r))) in TAUT A ) by Lm1; hence (p => (q => r)) => ((s => q) => (p => (s => r))) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm3: for A being QC-alphabet for p, q, r, s being Element of CQC-WFF A holds (p => q) => (((p => r) => s) => ((q => r) => s)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r, s being Element of CQC-WFF A holds (p => q) => (((p => r) => s) => ((q => r) => s)) in TAUT A let p, q, r, s be Element of CQC-WFF A; ::_thesis: (p => q) => (((p => r) => s) => ((q => r) => s)) in TAUT A ( ((q => r) => (p => r)) => (((p => r) => s) => ((q => r) => s)) in TAUT A & (((q => r) => (p => r)) => (((p => r) => s) => ((q => r) => s))) => ((p => q) => (((p => r) => s) => ((q => r) => s))) in TAUT A ) by Lm1, Th1; hence (p => q) => (((p => r) => s) => ((q => r) => s)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm4: for A being QC-alphabet for t, p, r, s, q being Element of CQC-WFF A holds (t => ((p => r) => s)) => ((p => q) => (t => ((q => r) => s))) in TAUT A proof let A be QC-alphabet ; ::_thesis: for t, p, r, s, q being Element of CQC-WFF A holds (t => ((p => r) => s)) => ((p => q) => (t => ((q => r) => s))) in TAUT A let t, p, r, s, q be Element of CQC-WFF A; ::_thesis: (t => ((p => r) => s)) => ((p => q) => (t => ((q => r) => s))) in TAUT A ( (p => q) => (((p => r) => s) => ((q => r) => s)) in TAUT A & ((p => q) => (((p => r) => s) => ((q => r) => s))) => ((t => ((p => r) => s)) => ((p => q) => (t => ((q => r) => s)))) in TAUT A ) by Lm2, Lm3; hence (t => ((p => r) => s)) => ((p => q) => (t => ((q => r) => s))) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm5: for A being QC-alphabet for p, q, r being Element of CQC-WFF A holds ((('not' p) => q) => r) => (p => r) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A holds ((('not' p) => q) => r) => (p => r) in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: ((('not' p) => q) => r) => (p => r) in TAUT A p => (('not' p) => q) in TAUT A by CQC_THE1:43; hence ((('not' p) => q) => r) => (p => r) in TAUT A by Th2; ::_thesis: verum end; Lm6: for A being QC-alphabet for p, r, s, q being Element of CQC-WFF A holds p => (((('not' p) => r) => s) => ((q => r) => s)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, r, s, q being Element of CQC-WFF A holds p => (((('not' p) => r) => s) => ((q => r) => s)) in TAUT A let p, r, s, q be Element of CQC-WFF A; ::_thesis: p => (((('not' p) => r) => s) => ((q => r) => s)) in TAUT A ( (('not' p) => q) => (((('not' p) => r) => s) => ((q => r) => s)) in TAUT A & ((('not' p) => q) => (((('not' p) => r) => s) => ((q => r) => s))) => (p => (((('not' p) => r) => s) => ((q => r) => s))) in TAUT A ) by Lm3, Lm5; hence p => (((('not' p) => r) => s) => ((q => r) => s)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm7: for A being QC-alphabet for q, p being Element of CQC-WFF A holds (q => ((('not' p) => p) => p)) => ((('not' p) => p) => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, p being Element of CQC-WFF A holds (q => ((('not' p) => p) => p)) => ((('not' p) => p) => p) in TAUT A let q, p be Element of CQC-WFF A; ::_thesis: (q => ((('not' p) => p) => p)) => ((('not' p) => p) => p) in TAUT A ( (('not' p) => p) => p in TAUT A & ((('not' p) => p) => p) => (((('not' ((('not' p) => p) => p)) => ((('not' p) => p) => p)) => ((('not' p) => p) => p)) => ((q => ((('not' p) => p) => p)) => ((('not' p) => p) => p))) in TAUT A ) by Lm6, CQC_THE1:42; then ( (('not' ((('not' p) => p) => p)) => ((('not' p) => p) => p)) => ((('not' p) => p) => p) in TAUT A & ((('not' ((('not' p) => p) => p)) => ((('not' p) => p) => p)) => ((('not' p) => p) => p)) => ((q => ((('not' p) => p) => p)) => ((('not' p) => p) => p)) in TAUT A ) by CQC_THE1:42, CQC_THE1:46; hence (q => ((('not' p) => p) => p)) => ((('not' p) => p) => p) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm8: for A being QC-alphabet for t, p being Element of CQC-WFF A holds t => ((('not' p) => p) => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for t, p being Element of CQC-WFF A holds t => ((('not' p) => p) => p) in TAUT A let t, p be Element of CQC-WFF A; ::_thesis: t => ((('not' p) => p) => p) in TAUT A ( (('not' t) => ((('not' p) => p) => p)) => ((('not' p) => p) => p) in TAUT A & ((('not' t) => ((('not' p) => p) => p)) => ((('not' p) => p) => p)) => (t => ((('not' p) => p) => p)) in TAUT A ) by Lm5, Lm7; hence t => ((('not' p) => p) => p) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm9: for A being QC-alphabet for p, q, t being Element of CQC-WFF A holds (('not' p) => q) => (t => ((q => p) => p)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, t being Element of CQC-WFF A holds (('not' p) => q) => (t => ((q => p) => p)) in TAUT A let p, q, t be Element of CQC-WFF A; ::_thesis: (('not' p) => q) => (t => ((q => p) => p)) in TAUT A ( t => ((('not' p) => p) => p) in TAUT A & (t => ((('not' p) => p) => p)) => ((('not' p) => q) => (t => ((q => p) => p))) in TAUT A ) by Lm4, Lm8; hence (('not' p) => q) => (t => ((q => p) => p)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm10: for A being QC-alphabet for t, q, p, r being Element of CQC-WFF A holds ((t => ((q => p) => p)) => r) => ((('not' p) => q) => r) in TAUT A proof let A be QC-alphabet ; ::_thesis: for t, q, p, r being Element of CQC-WFF A holds ((t => ((q => p) => p)) => r) => ((('not' p) => q) => r) in TAUT A let t, q, p, r be Element of CQC-WFF A; ::_thesis: ((t => ((q => p) => p)) => r) => ((('not' p) => q) => r) in TAUT A ( (('not' p) => q) => (t => ((q => p) => p)) in TAUT A & ((('not' p) => q) => (t => ((q => p) => p))) => (((t => ((q => p) => p)) => r) => ((('not' p) => q) => r)) in TAUT A ) by Lm9, Th1; hence ((t => ((q => p) => p)) => r) => ((('not' p) => q) => r) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm11: for A being QC-alphabet for p, q being Element of CQC-WFF A holds (('not' p) => q) => ((q => p) => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds (('not' p) => q) => ((q => p) => p) in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: (('not' p) => q) => ((q => p) => p) in TAUT A ( (('not' ((q => p) => p)) => ((q => p) => p)) => ((q => p) => p) in TAUT A & ((('not' ((q => p) => p)) => ((q => p) => p)) => ((q => p) => p)) => ((('not' p) => q) => ((q => p) => p)) in TAUT A ) by Lm10, CQC_THE1:42; hence (('not' p) => q) => ((q => p) => p) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm12: for A being QC-alphabet for p, q being Element of CQC-WFF A holds p => ((q => p) => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds p => ((q => p) => p) in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: p => ((q => p) => p) in TAUT A ( (('not' p) => q) => ((q => p) => p) in TAUT A & ((('not' p) => q) => ((q => p) => p)) => (p => ((q => p) => p)) in TAUT A ) by Lm5, Lm11; hence p => ((q => p) => p) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th5: :: LUKASI_1:5 for A being QC-alphabet for q, p being Element of CQC-WFF A holds q => (p => q) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, p being Element of CQC-WFF A holds q => (p => q) in TAUT A let q, p be Element of CQC-WFF A; ::_thesis: q => (p => q) in TAUT A ( q => ((('not' p) => q) => q) in TAUT A & (q => ((('not' p) => q) => q)) => ((p => (('not' p) => q)) => (q => (p => q))) in TAUT A ) by Lm2, Lm12; then ( p => (('not' p) => q) in TAUT A & (p => (('not' p) => q)) => (q => (p => q)) in TAUT A ) by CQC_THE1:43, CQC_THE1:46; hence q => (p => q) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th6: :: LUKASI_1:6 for A being QC-alphabet for p, q, r being Element of CQC-WFF A holds ((p => q) => r) => (q => r) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A holds ((p => q) => r) => (q => r) in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: ((p => q) => r) => (q => r) in TAUT A ( q => (p => q) in TAUT A & (q => (p => q)) => (((p => q) => r) => (q => r)) in TAUT A ) by Th1, Th5; hence ((p => q) => r) => (q => r) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th7: :: LUKASI_1:7 for A being QC-alphabet for q, p being Element of CQC-WFF A holds q => ((q => p) => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, p being Element of CQC-WFF A holds q => ((q => p) => p) in TAUT A let q, p be Element of CQC-WFF A; ::_thesis: q => ((q => p) => p) in TAUT A ( (('not' p) => q) => ((q => p) => p) in TAUT A & ((('not' p) => q) => ((q => p) => p)) => (q => ((q => p) => p)) in TAUT A ) by Lm11, Th6; hence q => ((q => p) => p) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th8: :: LUKASI_1:8 for A being QC-alphabet for s, q, p being Element of CQC-WFF A holds (s => (q => p)) => (q => (s => p)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for s, q, p being Element of CQC-WFF A holds (s => (q => p)) => (q => (s => p)) in TAUT A let s, q, p be Element of CQC-WFF A; ::_thesis: (s => (q => p)) => (q => (s => p)) in TAUT A ( q => ((q => p) => p) in TAUT A & (q => ((q => p) => p)) => ((s => (q => p)) => (q => (s => p))) in TAUT A ) by Lm2, Th7; hence (s => (q => p)) => (q => (s => p)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th9: :: LUKASI_1:9 for A being QC-alphabet for q, r, p being Element of CQC-WFF A holds (q => r) => ((p => q) => (p => r)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, r, p being Element of CQC-WFF A holds (q => r) => ((p => q) => (p => r)) in TAUT A let q, r, p be Element of CQC-WFF A; ::_thesis: (q => r) => ((p => q) => (p => r)) in TAUT A ( (p => q) => ((q => r) => (p => r)) in TAUT A & ((p => q) => ((q => r) => (p => r))) => ((q => r) => ((p => q) => (p => r))) in TAUT A ) by Th1, Th8; hence (q => r) => ((p => q) => (p => r)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm13: for A being QC-alphabet for q, s, p, r being Element of CQC-WFF A holds ((q => (s => p)) => r) => ((s => (q => p)) => r) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, s, p, r being Element of CQC-WFF A holds ((q => (s => p)) => r) => ((s => (q => p)) => r) in TAUT A let q, s, p, r be Element of CQC-WFF A; ::_thesis: ((q => (s => p)) => r) => ((s => (q => p)) => r) in TAUT A ( (s => (q => p)) => (q => (s => p)) in TAUT A & ((s => (q => p)) => (q => (s => p))) => (((q => (s => p)) => r) => ((s => (q => p)) => r)) in TAUT A ) by Th1, Th8; hence ((q => (s => p)) => r) => ((s => (q => p)) => r) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm14: for A being QC-alphabet for p, q being Element of CQC-WFF A holds ((p => q) => p) => p in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ((p => q) => p) => p in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: ((p => q) => p) => p in TAUT A ( (('not' p) => (p => q)) => (((p => q) => p) => p) in TAUT A & ((('not' p) => (p => q)) => (((p => q) => p) => p)) => ((p => (('not' p) => q)) => (((p => q) => p) => p)) in TAUT A ) by Lm11, Lm13; then ( p => (('not' p) => q) in TAUT A & (p => (('not' p) => q)) => (((p => q) => p) => p) in TAUT A ) by CQC_THE1:43, CQC_THE1:46; hence ((p => q) => p) => p in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm15: for A being QC-alphabet for p, r, s, q being Element of CQC-WFF A holds ((p => r) => s) => ((p => q) => ((q => r) => s)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, r, s, q being Element of CQC-WFF A holds ((p => r) => s) => ((p => q) => ((q => r) => s)) in TAUT A let p, r, s, q be Element of CQC-WFF A; ::_thesis: ((p => r) => s) => ((p => q) => ((q => r) => s)) in TAUT A ( (p => q) => (((p => r) => s) => ((q => r) => s)) in TAUT A & ((p => q) => (((p => r) => s) => ((q => r) => s))) => (((p => r) => s) => ((p => q) => ((q => r) => s))) in TAUT A ) by Lm3, Th8; hence ((p => r) => s) => ((p => q) => ((q => r) => s)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm16: for A being QC-alphabet for p, q, r being Element of CQC-WFF A holds ((p => q) => r) => ((r => p) => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A holds ((p => q) => r) => ((r => p) => p) in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: ((p => q) => r) => ((r => p) => p) in TAUT A ( ((p => q) => p) => p in TAUT A & (((p => q) => p) => p) => (((p => q) => r) => ((r => p) => p)) in TAUT A ) by Lm14, Lm15; hence ((p => q) => r) => ((r => p) => p) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm17: for A being QC-alphabet for r, p, s, q being Element of CQC-WFF A holds (((r => p) => p) => s) => (((p => q) => r) => s) in TAUT A proof let A be QC-alphabet ; ::_thesis: for r, p, s, q being Element of CQC-WFF A holds (((r => p) => p) => s) => (((p => q) => r) => s) in TAUT A let r, p, s, q be Element of CQC-WFF A; ::_thesis: (((r => p) => p) => s) => (((p => q) => r) => s) in TAUT A ( ((p => q) => r) => ((r => p) => p) in TAUT A & (((p => q) => r) => ((r => p) => p)) => ((((r => p) => p) => s) => (((p => q) => r) => s)) in TAUT A ) by Lm16, Th1; hence (((r => p) => p) => s) => (((p => q) => r) => s) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm18: for A being QC-alphabet for q, r, p being Element of CQC-WFF A holds ((q => r) => p) => ((q => p) => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, r, p being Element of CQC-WFF A holds ((q => r) => p) => ((q => p) => p) in TAUT A let q, r, p be Element of CQC-WFF A; ::_thesis: ((q => r) => p) => ((q => p) => p) in TAUT A ( ((p => q) => q) => ((q => p) => p) in TAUT A & (((p => q) => q) => ((q => p) => p)) => (((q => r) => p) => ((q => p) => p)) in TAUT A ) by Lm16, Lm17; hence ((q => r) => p) => ((q => p) => p) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th10: :: LUKASI_1:10 for A being QC-alphabet for q, r being Element of CQC-WFF A holds (q => (q => r)) => (q => r) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, r being Element of CQC-WFF A holds (q => (q => r)) => (q => r) in TAUT A let q, r be Element of CQC-WFF A; ::_thesis: (q => (q => r)) => (q => r) in TAUT A ( (q => r) => (q => r) in TAUT A & ((q => r) => (q => r)) => ((q => (q => r)) => (q => r)) in TAUT A ) by Lm18, Th4; hence (q => (q => r)) => (q => r) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm19: for A being QC-alphabet for q, s, r, p being Element of CQC-WFF A holds (q => s) => (((q => r) => p) => ((s => p) => p)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, s, r, p being Element of CQC-WFF A holds (q => s) => (((q => r) => p) => ((s => p) => p)) in TAUT A let q, s, r, p be Element of CQC-WFF A; ::_thesis: (q => s) => (((q => r) => p) => ((s => p) => p)) in TAUT A ( ((q => r) => p) => ((q => p) => p) in TAUT A & (((q => r) => p) => ((q => p) => p)) => ((q => s) => (((q => r) => p) => ((s => p) => p))) in TAUT A ) by Lm4, Lm18; hence (q => s) => (((q => r) => p) => ((s => p) => p)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm20: for A being QC-alphabet for q, r, p, s being Element of CQC-WFF A holds ((q => r) => p) => ((q => s) => ((s => p) => p)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, r, p, s being Element of CQC-WFF A holds ((q => r) => p) => ((q => s) => ((s => p) => p)) in TAUT A let q, r, p, s be Element of CQC-WFF A; ::_thesis: ((q => r) => p) => ((q => s) => ((s => p) => p)) in TAUT A ( (q => s) => (((q => r) => p) => ((s => p) => p)) in TAUT A & ((q => s) => (((q => r) => p) => ((s => p) => p))) => (((q => r) => p) => ((q => s) => ((s => p) => p))) in TAUT A ) by Lm19, Th8; hence ((q => r) => p) => ((q => s) => ((s => p) => p)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm21: for A being QC-alphabet for q, s, p, r being Element of CQC-WFF A holds (q => s) => ((s => (p => (q => r))) => (p => (q => r))) in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, s, p, r being Element of CQC-WFF A holds (q => s) => ((s => (p => (q => r))) => (p => (q => r))) in TAUT A let q, s, p, r be Element of CQC-WFF A; ::_thesis: (q => s) => ((s => (p => (q => r))) => (p => (q => r))) in TAUT A ( (q => r) => (p => (q => r)) in TAUT A & ((q => r) => (p => (q => r))) => ((q => s) => ((s => (p => (q => r))) => (p => (q => r)))) in TAUT A ) by Lm20, Th5; hence (q => s) => ((s => (p => (q => r))) => (p => (q => r))) in TAUT A by CQC_THE1:46; ::_thesis: verum end; Lm22: for A being QC-alphabet for s, p, q, r being Element of CQC-WFF A holds (s => (p => (q => r))) => ((q => s) => (p => (q => r))) in TAUT A proof let A be QC-alphabet ; ::_thesis: for s, p, q, r being Element of CQC-WFF A holds (s => (p => (q => r))) => ((q => s) => (p => (q => r))) in TAUT A let s, p, q, r be Element of CQC-WFF A; ::_thesis: (s => (p => (q => r))) => ((q => s) => (p => (q => r))) in TAUT A ( (q => s) => ((s => (p => (q => r))) => (p => (q => r))) in TAUT A & ((q => s) => ((s => (p => (q => r))) => (p => (q => r)))) => ((s => (p => (q => r))) => ((q => s) => (p => (q => r)))) in TAUT A ) by Lm21, Th8; hence (s => (p => (q => r))) => ((q => s) => (p => (q => r))) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th11: :: LUKASI_1:11 for A being QC-alphabet for p, q, r being Element of CQC-WFF A holds (p => (q => r)) => ((p => q) => (p => r)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A holds (p => (q => r)) => ((p => q) => (p => r)) in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: (p => (q => r)) => ((p => q) => (p => r)) in TAUT A ( (q => r) => ((p => q) => (p => r)) in TAUT A & ((q => r) => ((p => q) => (p => r))) => ((p => (q => r)) => ((p => q) => (p => r))) in TAUT A ) by Lm22, Th9; hence (p => (q => r)) => ((p => q) => (p => r)) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th12: :: LUKASI_1:12 for A being QC-alphabet for p being Element of CQC-WFF A holds ('not' (VERUM A)) => p in TAUT A proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds ('not' (VERUM A)) => p in TAUT A let p be Element of CQC-WFF A; ::_thesis: ('not' (VERUM A)) => p in TAUT A (VERUM A) => (('not' (VERUM A)) => p) in TAUT A by CQC_THE1:43; hence ('not' (VERUM A)) => p in TAUT A by CQC_THE1:41, CQC_THE1:46; ::_thesis: verum end; theorem Th13: :: LUKASI_1:13 for A being QC-alphabet for q, p being Element of CQC-WFF A st q in TAUT A holds p => q in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, p being Element of CQC-WFF A st q in TAUT A holds p => q in TAUT A let q, p be Element of CQC-WFF A; ::_thesis: ( q in TAUT A implies p => q in TAUT A ) q => (p => q) in TAUT A by Th5; hence ( q in TAUT A implies p => q in TAUT A ) by CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:14 for A being QC-alphabet for p, q being Element of CQC-WFF A st p in TAUT A holds (p => q) => q in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A st p in TAUT A holds (p => q) => q in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: ( p in TAUT A implies (p => q) => q in TAUT A ) assume A1: p in TAUT A ; ::_thesis: (p => q) => q in TAUT A p => ((p => q) => q) in TAUT A by Th7; hence (p => q) => q in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; theorem Th15: :: LUKASI_1:15 for A being QC-alphabet for s, q, p being Element of CQC-WFF A st s => (q => p) in TAUT A holds q => (s => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for s, q, p being Element of CQC-WFF A st s => (q => p) in TAUT A holds q => (s => p) in TAUT A let s, q, p be Element of CQC-WFF A; ::_thesis: ( s => (q => p) in TAUT A implies q => (s => p) in TAUT A ) assume A1: s => (q => p) in TAUT A ; ::_thesis: q => (s => p) in TAUT A (s => (q => p)) => (q => (s => p)) in TAUT A by Th8; hence q => (s => p) in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; theorem Th16: :: LUKASI_1:16 for A being QC-alphabet for s, q, p being Element of CQC-WFF A st s => (q => p) in TAUT A & q in TAUT A holds s => p in TAUT A proof let A be QC-alphabet ; ::_thesis: for s, q, p being Element of CQC-WFF A st s => (q => p) in TAUT A & q in TAUT A holds s => p in TAUT A let s, q, p be Element of CQC-WFF A; ::_thesis: ( s => (q => p) in TAUT A & q in TAUT A implies s => p in TAUT A ) assume s => (q => p) in TAUT A ; ::_thesis: ( not q in TAUT A or s => p in TAUT A ) then q => (s => p) in TAUT A by Th15; hence ( not q in TAUT A or s => p in TAUT A ) by CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:17 for A being QC-alphabet for s, q, p being Element of CQC-WFF A st s => (q => p) in TAUT A & q in TAUT A & s in TAUT A holds p in TAUT A proof let A be QC-alphabet ; ::_thesis: for s, q, p being Element of CQC-WFF A st s => (q => p) in TAUT A & q in TAUT A & s in TAUT A holds p in TAUT A let s, q, p be Element of CQC-WFF A; ::_thesis: ( s => (q => p) in TAUT A & q in TAUT A & s in TAUT A implies p in TAUT A ) assume ( s => (q => p) in TAUT A & q in TAUT A ) ; ::_thesis: ( not s in TAUT A or p in TAUT A ) then s => p in TAUT A by Th16; hence ( not s in TAUT A or p in TAUT A ) by CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:18 for A being QC-alphabet for q, r being Element of CQC-WFF A st q => (q => r) in TAUT A holds q => r in TAUT A proof let A be QC-alphabet ; ::_thesis: for q, r being Element of CQC-WFF A st q => (q => r) in TAUT A holds q => r in TAUT A let q, r be Element of CQC-WFF A; ::_thesis: ( q => (q => r) in TAUT A implies q => r in TAUT A ) (q => (q => r)) => (q => r) in TAUT A by Th10; hence ( q => (q => r) in TAUT A implies q => r in TAUT A ) by CQC_THE1:46; ::_thesis: verum end; theorem Th19: :: LUKASI_1:19 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A holds (p => q) => (p => r) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A holds (p => q) => (p => r) in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => (q => r) in TAUT A implies (p => q) => (p => r) in TAUT A ) assume A1: p => (q => r) in TAUT A ; ::_thesis: (p => q) => (p => r) in TAUT A (p => (q => r)) => ((p => q) => (p => r)) in TAUT A by Th11; hence (p => q) => (p => r) in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; theorem Th20: :: LUKASI_1:20 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A & p => q in TAUT A holds p => r in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A & p => q in TAUT A holds p => r in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => (q => r) in TAUT A & p => q in TAUT A implies p => r in TAUT A ) assume p => (q => r) in TAUT A ; ::_thesis: ( not p => q in TAUT A or p => r in TAUT A ) then (p => q) => (p => r) in TAUT A by Th19; hence ( not p => q in TAUT A or p => r in TAUT A ) by CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:21 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A & p => q in TAUT A & p in TAUT A holds r in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => (q => r) in TAUT A & p => q in TAUT A & p in TAUT A holds r in TAUT A let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => (q => r) in TAUT A & p => q in TAUT A & p in TAUT A implies r in TAUT A ) assume ( p => (q => r) in TAUT A & p => q in TAUT A ) ; ::_thesis: ( not p in TAUT A or r in TAUT A ) then p => r in TAUT A by Th20; hence ( not p in TAUT A or r in TAUT A ) by CQC_THE1:46; ::_thesis: verum end; theorem Th22: :: LUKASI_1:22 for A being QC-alphabet for p, q, r, s being Element of CQC-WFF A st p => (q => r) in TAUT A & p => (r => s) in TAUT A holds p => (q => s) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q, r, s being Element of CQC-WFF A st p => (q => r) in TAUT A & p => (r => s) in TAUT A holds p => (q => s) in TAUT A let p, q, r, s be Element of CQC-WFF A; ::_thesis: ( p => (q => r) in TAUT A & p => (r => s) in TAUT A implies p => (q => s) in TAUT A ) assume that A1: p => (q => r) in TAUT A and A2: p => (r => s) in TAUT A ; ::_thesis: p => (q => s) in TAUT A p => ((q => r) => ((r => s) => (q => s))) in TAUT A by Th1, Th13; then p => ((r => s) => (q => s)) in TAUT A by A1, Th20; hence p => (q => s) in TAUT A by A2, Th20; ::_thesis: verum end; theorem :: LUKASI_1:23 for A being QC-alphabet for p being Element of CQC-WFF A holds p => (VERUM A) in TAUT A by Th13, CQC_THE1:41; Lm23: for A being QC-alphabet for p being Element of CQC-WFF A holds ('not' p) => (p => ('not' (VERUM A))) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds ('not' p) => (p => ('not' (VERUM A))) in TAUT A let p be Element of CQC-WFF A; ::_thesis: ('not' p) => (p => ('not' (VERUM A))) in TAUT A p => (('not' p) => ('not' (VERUM A))) in TAUT A by CQC_THE1:43; hence ('not' p) => (p => ('not' (VERUM A))) in TAUT A by Th15; ::_thesis: verum end; Lm24: for A being QC-alphabet for p being Element of CQC-WFF A holds (('not' p) => ('not' (VERUM A))) => p in TAUT A proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds (('not' p) => ('not' (VERUM A))) => p in TAUT A let p be Element of CQC-WFF A; ::_thesis: (('not' p) => ('not' (VERUM A))) => p in TAUT A ( ('not' p) => (('not' (VERUM A)) => p) in TAUT A & (('not' p) => (('not' (VERUM A)) => p)) => ((('not' p) => ('not' (VERUM A))) => (('not' p) => p)) in TAUT A ) by Th11, Th12, Th13; then A1: (('not' p) => ('not' (VERUM A))) => (('not' p) => p) in TAUT A by CQC_THE1:46; (('not' p) => p) => p in TAUT A by CQC_THE1:42; hence (('not' p) => ('not' (VERUM A))) => p in TAUT A by A1, Th3; ::_thesis: verum end; theorem Th24: :: LUKASI_1:24 for A being QC-alphabet for p, q being Element of CQC-WFF A holds (('not' p) => ('not' q)) => (q => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds (('not' p) => ('not' q)) => (q => p) in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: (('not' p) => ('not' q)) => (q => p) in TAUT A ( q => (('not' q) => ('not' (VERUM A))) in TAUT A & (('not' q) => ('not' (VERUM A))) => ((('not' p) => ('not' q)) => (('not' p) => ('not' (VERUM A)))) in TAUT A ) by Th9, CQC_THE1:43; then A1: q => ((('not' p) => ('not' q)) => (('not' p) => ('not' (VERUM A)))) in TAUT A by Th3; q => ((('not' p) => ('not' (VERUM A))) => p) in TAUT A by Lm24, Th13; then q => ((('not' p) => ('not' q)) => p) in TAUT A by A1, Th22; hence (('not' p) => ('not' q)) => (q => p) in TAUT A by Th15; ::_thesis: verum end; theorem Th25: :: LUKASI_1:25 for A being QC-alphabet for p being Element of CQC-WFF A holds ('not' ('not' p)) => p in TAUT A proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds ('not' ('not' p)) => p in TAUT A let p be Element of CQC-WFF A; ::_thesis: ('not' ('not' p)) => p in TAUT A ( ('not' ('not' p)) => (('not' p) => ('not' (VERUM A))) in TAUT A & (('not' p) => ('not' (VERUM A))) => ((VERUM A) => p) in TAUT A ) by Lm23, Th24; then ('not' ('not' p)) => ((VERUM A) => p) in TAUT A by Th3; then (VERUM A) => (('not' ('not' p)) => p) in TAUT A by Th15; hence ('not' ('not' p)) => p in TAUT A by CQC_THE1:41, CQC_THE1:46; ::_thesis: verum end; Lm25: now__::_thesis:_for_A_being_QC-alphabet_ for_p_being_Element_of_CQC-WFF_A_holds_(p_=>_('not'_(VERUM_A)))_=>_('not'_p)_in_TAUT_A let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds (p => ('not' (VERUM A))) => ('not' p) in TAUT A let p be Element of CQC-WFF A; ::_thesis: (p => ('not' (VERUM A))) => ('not' p) in TAUT A ('not' ('not' p)) => p in TAUT A by Th25; then A1: (p => ('not' (VERUM A))) => (('not' ('not' p)) => ('not' (VERUM A))) in TAUT A by Th2; (('not' ('not' p)) => ('not' (VERUM A))) => ('not' p) in TAUT A by Lm24; hence (p => ('not' (VERUM A))) => ('not' p) in TAUT A by A1, Th3; ::_thesis: verum end; theorem Th26: :: LUKASI_1:26 for A being QC-alphabet for p, q being Element of CQC-WFF A holds (p => q) => (('not' q) => ('not' p)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds (p => q) => (('not' q) => ('not' p)) in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: (p => q) => (('not' q) => ('not' p)) in TAUT A ( ('not' q) => (q => ('not' (VERUM A))) in TAUT A & (q => ('not' (VERUM A))) => ((p => q) => (p => ('not' (VERUM A)))) in TAUT A ) by Lm23, Th9; then A1: ('not' q) => ((p => q) => (p => ('not' (VERUM A)))) in TAUT A by Th3; ('not' q) => ((p => ('not' (VERUM A))) => ('not' p)) in TAUT A by Lm25, Th13; then ('not' q) => ((p => q) => ('not' p)) in TAUT A by A1, Th22; hence (p => q) => (('not' q) => ('not' p)) in TAUT A by Th15; ::_thesis: verum end; theorem Th27: :: LUKASI_1:27 for A being QC-alphabet for p being Element of CQC-WFF A holds p => ('not' ('not' p)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds p => ('not' ('not' p)) in TAUT A let p be Element of CQC-WFF A; ::_thesis: p => ('not' ('not' p)) in TAUT A ( ((VERUM A) => p) => (('not' p) => ('not' (VERUM A))) in TAUT A & (('not' p) => ('not' (VERUM A))) => ('not' ('not' p)) in TAUT A ) by Lm25, Th26; then A1: ((VERUM A) => p) => ('not' ('not' p)) in TAUT A by Th3; p => ((VERUM A) => p) in TAUT A by Th5; hence p => ('not' ('not' p)) in TAUT A by A1, Th3; ::_thesis: verum end; theorem Th28: :: LUKASI_1:28 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ( (('not' ('not' p)) => q) => (p => q) in TAUT A & (p => q) => (('not' ('not' p)) => q) in TAUT A ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ( (('not' ('not' p)) => q) => (p => q) in TAUT A & (p => q) => (('not' ('not' p)) => q) in TAUT A ) let p, q be Element of CQC-WFF A; ::_thesis: ( (('not' ('not' p)) => q) => (p => q) in TAUT A & (p => q) => (('not' ('not' p)) => q) in TAUT A ) p => ('not' ('not' p)) in TAUT A by Th27; hence (('not' ('not' p)) => q) => (p => q) in TAUT A by Th2; ::_thesis: (p => q) => (('not' ('not' p)) => q) in TAUT A ('not' ('not' p)) => p in TAUT A by Th25; hence (p => q) => (('not' ('not' p)) => q) in TAUT A by Th2; ::_thesis: verum end; theorem Th29: :: LUKASI_1:29 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ( (p => ('not' ('not' q))) => (p => q) in TAUT A & (p => q) => (p => ('not' ('not' q))) in TAUT A ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ( (p => ('not' ('not' q))) => (p => q) in TAUT A & (p => q) => (p => ('not' ('not' q))) in TAUT A ) let p, q be Element of CQC-WFF A; ::_thesis: ( (p => ('not' ('not' q))) => (p => q) in TAUT A & (p => q) => (p => ('not' ('not' q))) in TAUT A ) ( (p => (('not' ('not' q)) => q)) => ((p => ('not' ('not' q))) => (p => q)) in TAUT A & p => (('not' ('not' q)) => q) in TAUT A ) by Th11, Th13, Th25; hence (p => ('not' ('not' q))) => (p => q) in TAUT A by CQC_THE1:46; ::_thesis: (p => q) => (p => ('not' ('not' q))) in TAUT A ( (p => (q => ('not' ('not' q)))) => ((p => q) => (p => ('not' ('not' q)))) in TAUT A & p => (q => ('not' ('not' q))) in TAUT A ) by Th11, Th13, Th27; hence (p => q) => (p => ('not' ('not' q))) in TAUT A by CQC_THE1:46; ::_thesis: verum end; theorem Th30: :: LUKASI_1:30 for A being QC-alphabet for p, q being Element of CQC-WFF A holds (p => ('not' q)) => (q => ('not' p)) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds (p => ('not' q)) => (q => ('not' p)) in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: (p => ('not' q)) => (q => ('not' p)) in TAUT A ( (p => ('not' q)) => (('not' ('not' q)) => ('not' p)) in TAUT A & (('not' ('not' q)) => ('not' p)) => (q => ('not' p)) in TAUT A ) by Th26, Th28; hence (p => ('not' q)) => (q => ('not' p)) in TAUT A by Th3; ::_thesis: verum end; theorem Th31: :: LUKASI_1:31 for A being QC-alphabet for p, q being Element of CQC-WFF A holds (('not' p) => q) => (('not' q) => p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds (('not' p) => q) => (('not' q) => p) in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: (('not' p) => q) => (('not' q) => p) in TAUT A ( (('not' p) => q) => (('not' q) => ('not' ('not' p))) in TAUT A & (('not' q) => ('not' ('not' p))) => (('not' q) => p) in TAUT A ) by Th26, Th29; hence (('not' p) => q) => (('not' q) => p) in TAUT A by Th3; ::_thesis: verum end; theorem :: LUKASI_1:32 for A being QC-alphabet for p being Element of CQC-WFF A holds (p => ('not' p)) => ('not' p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds (p => ('not' p)) => ('not' p) in TAUT A let p be Element of CQC-WFF A; ::_thesis: (p => ('not' p)) => ('not' p) in TAUT A ( (('not' ('not' p)) => ('not' p)) => ('not' p) in TAUT A & (p => ('not' p)) => (('not' ('not' p)) => ('not' p)) in TAUT A ) by Th28, CQC_THE1:42; hence (p => ('not' p)) => ('not' p) in TAUT A by Th3; ::_thesis: verum end; theorem :: LUKASI_1:33 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ('not' p) => (p => q) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ('not' p) => (p => q) in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: ('not' p) => (p => q) in TAUT A ( ('not' p) => (('not' ('not' p)) => q) in TAUT A & (('not' ('not' p)) => q) => (p => q) in TAUT A ) by Th28, CQC_THE1:43; hence ('not' p) => (p => q) in TAUT A by Th3; ::_thesis: verum end; theorem Th34: :: LUKASI_1:34 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ( p => q in TAUT A iff ('not' q) => ('not' p) in TAUT A ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ( p => q in TAUT A iff ('not' q) => ('not' p) in TAUT A ) let p, q be Element of CQC-WFF A; ::_thesis: ( p => q in TAUT A iff ('not' q) => ('not' p) in TAUT A ) (p => q) => (('not' q) => ('not' p)) in TAUT A by Th26; hence ( p => q in TAUT A implies ('not' q) => ('not' p) in TAUT A ) by CQC_THE1:46; ::_thesis: ( ('not' q) => ('not' p) in TAUT A implies p => q in TAUT A ) (('not' q) => ('not' p)) => (p => q) in TAUT A by Th24; hence ( ('not' q) => ('not' p) in TAUT A implies p => q in TAUT A ) by CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:35 for A being QC-alphabet for p, q being Element of CQC-WFF A st ('not' p) => ('not' q) in TAUT A holds q => p in TAUT A by Th34; theorem :: LUKASI_1:36 for A being QC-alphabet for p being Element of CQC-WFF A holds ( p in TAUT A iff 'not' ('not' p) in TAUT A ) proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds ( p in TAUT A iff 'not' ('not' p) in TAUT A ) let p be Element of CQC-WFF A; ::_thesis: ( p in TAUT A iff 'not' ('not' p) in TAUT A ) thus ( p in TAUT A implies 'not' ('not' p) in TAUT A ) ::_thesis: ( 'not' ('not' p) in TAUT A implies p in TAUT A ) proof assume A1: p in TAUT A ; ::_thesis: 'not' ('not' p) in TAUT A p => ('not' ('not' p)) in TAUT A by Th27; hence 'not' ('not' p) in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; assume A2: 'not' ('not' p) in TAUT A ; ::_thesis: p in TAUT A ('not' ('not' p)) => p in TAUT A by Th25; hence p in TAUT A by A2, CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:37 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ( p => q in TAUT A iff p => ('not' ('not' q)) in TAUT A ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ( p => q in TAUT A iff p => ('not' ('not' q)) in TAUT A ) let p, q be Element of CQC-WFF A; ::_thesis: ( p => q in TAUT A iff p => ('not' ('not' q)) in TAUT A ) thus ( p => q in TAUT A implies p => ('not' ('not' q)) in TAUT A ) ::_thesis: ( p => ('not' ('not' q)) in TAUT A implies p => q in TAUT A ) proof assume A1: p => q in TAUT A ; ::_thesis: p => ('not' ('not' q)) in TAUT A (p => q) => (p => ('not' ('not' q))) in TAUT A by Th29; hence p => ('not' ('not' q)) in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; assume A2: p => ('not' ('not' q)) in TAUT A ; ::_thesis: p => q in TAUT A (p => ('not' ('not' q))) => (p => q) in TAUT A by Th29; hence p => q in TAUT A by A2, CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:38 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ( p => q in TAUT A iff ('not' ('not' p)) => q in TAUT A ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ( p => q in TAUT A iff ('not' ('not' p)) => q in TAUT A ) let p, q be Element of CQC-WFF A; ::_thesis: ( p => q in TAUT A iff ('not' ('not' p)) => q in TAUT A ) thus ( p => q in TAUT A implies ('not' ('not' p)) => q in TAUT A ) ::_thesis: ( ('not' ('not' p)) => q in TAUT A implies p => q in TAUT A ) proof assume A1: p => q in TAUT A ; ::_thesis: ('not' ('not' p)) => q in TAUT A (p => q) => (('not' ('not' p)) => q) in TAUT A by Th28; hence ('not' ('not' p)) => q in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; assume A2: ('not' ('not' p)) => q in TAUT A ; ::_thesis: p => q in TAUT A (('not' ('not' p)) => q) => (p => q) in TAUT A by Th28; hence p => q in TAUT A by A2, CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:39 for A being QC-alphabet for p, q being Element of CQC-WFF A st p => ('not' q) in TAUT A holds q => ('not' p) in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A st p => ('not' q) in TAUT A holds q => ('not' p) in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: ( p => ('not' q) in TAUT A implies q => ('not' p) in TAUT A ) assume A1: p => ('not' q) in TAUT A ; ::_thesis: q => ('not' p) in TAUT A (p => ('not' q)) => (q => ('not' p)) in TAUT A by Th30; hence q => ('not' p) in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; theorem :: LUKASI_1:40 for A being QC-alphabet for p, q being Element of CQC-WFF A st ('not' p) => q in TAUT A holds ('not' q) => p in TAUT A proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A st ('not' p) => q in TAUT A holds ('not' q) => p in TAUT A let p, q be Element of CQC-WFF A; ::_thesis: ( ('not' p) => q in TAUT A implies ('not' q) => p in TAUT A ) assume A1: ('not' p) => q in TAUT A ; ::_thesis: ('not' q) => p in TAUT A (('not' p) => q) => (('not' q) => p) in TAUT A by Th31; hence ('not' q) => p in TAUT A by A1, CQC_THE1:46; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q, r be Element of CQC-WFF A; clusterK170(A,(p => q),((q => r) => (p => r))) -> valid ; coherence (p => q) => ((q => r) => (p => r)) is valid proof thus (p => q) => ((q => r) => (p => r)) in TAUT A by Th1; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:41 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => q is valid holds (q => r) => (p => r) is valid proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => q is valid holds (q => r) => (p => r) is valid let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => q is valid implies (q => r) => (p => r) is valid ) assume A1: p => q is valid ; ::_thesis: (q => r) => (p => r) is valid (p => q) => ((q => r) => (p => r)) is valid ; hence (q => r) => (p => r) is valid by A1, CQC_THE1:65; ::_thesis: verum end; theorem Th42: :: LUKASI_1:42 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => q is valid & q => r is valid holds p => r is valid proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => q is valid & q => r is valid holds p => r is valid let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => q is valid & q => r is valid implies p => r is valid ) assume ( p => q is valid & q => r is valid ) ; ::_thesis: p => r is valid then ( p => q in TAUT A & q => r in TAUT A ) by CQC_THE1:def_10; hence p => r in TAUT A by Th3; :: according to CQC_THE1:def_10 ::_thesis: verum end; registration let A be QC-alphabet ; let p be Element of CQC-WFF A; clusterK170(A,p,p) -> valid ; coherence p => p is valid proof thus p => p in TAUT A by Th4; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; registration let A be QC-alphabet ; let p, q be Element of CQC-WFF A; clusterK170(A,p,(q => p)) -> valid ; coherence p => (q => p) is valid proof thus p => (q => p) in TAUT A by Th5; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:43 for A being QC-alphabet for p, q being Element of CQC-WFF A st p is valid holds q => p is valid proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A st p is valid holds q => p is valid let p, q be Element of CQC-WFF A; ::_thesis: ( p is valid implies q => p is valid ) assume p is valid ; ::_thesis: q => p is valid then p in TAUT A by CQC_THE1:def_10; hence q => p in TAUT A by Th13; :: according to CQC_THE1:def_10 ::_thesis: verum end; registration let A be QC-alphabet ; let p, q, s be Element of CQC-WFF A; clusterK170(A,(s => (q => p)),(q => (s => p))) -> valid ; coherence (s => (q => p)) => (q => (s => p)) is valid proof thus (s => (q => p)) => (q => (s => p)) in TAUT A by Th8; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem Th44: :: LUKASI_1:44 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => (q => r) is valid holds q => (p => r) is valid proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => (q => r) is valid holds q => (p => r) is valid let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => (q => r) is valid implies q => (p => r) is valid ) assume p => (q => r) is valid ; ::_thesis: q => (p => r) is valid then p => (q => r) in TAUT A by CQC_THE1:def_10; hence q => (p => r) in TAUT A by Th15; :: according to CQC_THE1:def_10 ::_thesis: verum end; theorem :: LUKASI_1:45 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => (q => r) is valid & q is valid holds p => r is valid proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => (q => r) is valid & q is valid holds p => r is valid let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => (q => r) is valid & q is valid implies p => r is valid ) assume p => (q => r) is valid ; ::_thesis: ( not q is valid or p => r is valid ) then q => (p => r) is valid by Th44; hence ( not q is valid or p => r is valid ) by CQC_THE1:65; ::_thesis: verum end; theorem :: LUKASI_1:46 for A being QC-alphabet for p being Element of CQC-WFF A holds ( p => (VERUM A) is valid & ('not' (VERUM A)) => p is valid ) proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds ( p => (VERUM A) is valid & ('not' (VERUM A)) => p is valid ) let p be Element of CQC-WFF A; ::_thesis: ( p => (VERUM A) is valid & ('not' (VERUM A)) => p is valid ) thus p => (VERUM A) in TAUT A by Th13, CQC_THE1:41; :: according to CQC_THE1:def_10 ::_thesis: ('not' (VERUM A)) => p is valid thus ('not' (VERUM A)) => p in TAUT A by Th12; :: according to CQC_THE1:def_10 ::_thesis: verum end; registration let A be QC-alphabet ; let p, q be Element of CQC-WFF A; clusterK170(A,p,((p => q) => q)) -> valid ; coherence p => ((p => q) => q) is valid proof thus p => ((p => q) => q) in TAUT A by Th7; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; registration let A be QC-alphabet ; let q, r be Element of CQC-WFF A; clusterK170(A,(q => (q => r)),(q => r)) -> valid ; coherence (q => (q => r)) => (q => r) is valid proof thus (q => (q => r)) => (q => r) in TAUT A by Th10; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:47 for A being QC-alphabet for q, r being Element of CQC-WFF A st q => (q => r) is valid holds q => r is valid proof let A be QC-alphabet ; ::_thesis: for q, r being Element of CQC-WFF A st q => (q => r) is valid holds q => r is valid let q, r be Element of CQC-WFF A; ::_thesis: ( q => (q => r) is valid implies q => r is valid ) assume A1: q => (q => r) is valid ; ::_thesis: q => r is valid (q => (q => r)) => (q => r) is valid ; hence q => r is valid by A1, CQC_THE1:65; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q, r be Element of CQC-WFF A; clusterK170(A,(p => (q => r)),((p => q) => (p => r))) -> valid ; coherence (p => (q => r)) => ((p => q) => (p => r)) is valid proof thus (p => (q => r)) => ((p => q) => (p => r)) in TAUT A by Th11; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem Th48: :: LUKASI_1:48 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => (q => r) is valid holds (p => q) => (p => r) is valid proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => (q => r) is valid holds (p => q) => (p => r) is valid let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => (q => r) is valid implies (p => q) => (p => r) is valid ) assume A1: p => (q => r) is valid ; ::_thesis: (p => q) => (p => r) is valid (p => (q => r)) => ((p => q) => (p => r)) is valid ; hence (p => q) => (p => r) is valid by A1, CQC_THE1:65; ::_thesis: verum end; theorem :: LUKASI_1:49 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => (q => r) is valid & p => q is valid holds p => r is valid proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => (q => r) is valid & p => q is valid holds p => r is valid let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => (q => r) is valid & p => q is valid implies p => r is valid ) assume that A1: p => (q => r) is valid and A2: p => q is valid ; ::_thesis: p => r is valid (p => q) => (p => r) is valid by A1, Th48; hence p => r is valid by A2, CQC_THE1:65; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q, r be Element of CQC-WFF A; clusterK170(A,((p => q) => r),(q => r)) -> valid ; coherence ((p => q) => r) => (q => r) is valid proof thus ((p => q) => r) => (q => r) in TAUT A by Th6; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:50 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st (p => q) => r is valid holds q => r is valid proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st (p => q) => r is valid holds q => r is valid let p, q, r be Element of CQC-WFF A; ::_thesis: ( (p => q) => r is valid implies q => r is valid ) assume A1: (p => q) => r is valid ; ::_thesis: q => r is valid ((p => q) => r) => (q => r) is valid ; hence q => r is valid by A1, CQC_THE1:65; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q, r be Element of CQC-WFF A; clusterK170(A,(p => q),((r => p) => (r => q))) -> valid ; coherence (p => q) => ((r => p) => (r => q)) is valid proof thus (p => q) => ((r => p) => (r => q)) in TAUT A by Th9; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:51 for A being QC-alphabet for p, q, r being Element of CQC-WFF A st p => q is valid holds (r => p) => (r => q) is valid proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p => q is valid holds (r => p) => (r => q) is valid let p, q, r be Element of CQC-WFF A; ::_thesis: ( p => q is valid implies (r => p) => (r => q) is valid ) assume A1: p => q is valid ; ::_thesis: (r => p) => (r => q) is valid (p => q) => ((r => p) => (r => q)) is valid ; hence (r => p) => (r => q) is valid by A1, CQC_THE1:65; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q be Element of CQC-WFF A; clusterK170(A,(p => q),(('not' q) => ('not' p))) -> valid ; coherence (p => q) => (('not' q) => ('not' p)) is valid proof thus (p => q) => (('not' q) => ('not' p)) in TAUT A by Th26; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; registration let A be QC-alphabet ; let p, q be Element of CQC-WFF A; clusterK170(A,(('not' p) => ('not' q)),(q => p)) -> valid ; coherence (('not' p) => ('not' q)) => (q => p) is valid proof thus (('not' p) => ('not' q)) => (q => p) in TAUT A by Th24; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:52 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ( ('not' p) => ('not' q) is valid iff q => p is valid ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ( ('not' p) => ('not' q) is valid iff q => p is valid ) let p, q be Element of CQC-WFF A; ::_thesis: ( ('not' p) => ('not' q) is valid iff q => p is valid ) thus ( ('not' p) => ('not' q) is valid implies q => p is valid ) ::_thesis: ( q => p is valid implies ('not' p) => ('not' q) is valid ) proof assume A1: ('not' p) => ('not' q) is valid ; ::_thesis: q => p is valid (('not' p) => ('not' q)) => (q => p) is valid ; hence q => p is valid by A1, CQC_THE1:65; ::_thesis: verum end; assume A2: q => p is valid ; ::_thesis: ('not' p) => ('not' q) is valid (q => p) => (('not' p) => ('not' q)) is valid ; hence ('not' p) => ('not' q) is valid by A2, CQC_THE1:65; ::_thesis: verum end; registration let A be QC-alphabet ; let p be Element of CQC-WFF A; clusterK170(A,p,('not' ('not' p))) -> valid ; coherence p => ('not' ('not' p)) is valid proof thus p => ('not' ('not' p)) in TAUT A by Th27; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; registration let A be QC-alphabet ; let p be Element of CQC-WFF A; clusterK170(A,('not' ('not' p)),p) -> valid ; coherence ('not' ('not' p)) => p is valid proof thus ('not' ('not' p)) => p in TAUT A by Th25; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:53 for A being QC-alphabet for p being Element of CQC-WFF A holds ( 'not' ('not' p) is valid iff p is valid ) proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds ( 'not' ('not' p) is valid iff p is valid ) let p be Element of CQC-WFF A; ::_thesis: ( 'not' ('not' p) is valid iff p is valid ) thus ( 'not' ('not' p) is valid implies p is valid ) ::_thesis: ( p is valid implies 'not' ('not' p) is valid ) proof assume A1: 'not' ('not' p) is valid ; ::_thesis: p is valid ('not' ('not' p)) => p is valid ; hence p is valid by A1, CQC_THE1:65; ::_thesis: verum end; assume A2: p is valid ; ::_thesis: 'not' ('not' p) is valid p => ('not' ('not' p)) is valid ; hence 'not' ('not' p) is valid by A2, CQC_THE1:65; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q be Element of CQC-WFF A; clusterK170(A,(('not' ('not' p)) => q),(p => q)) -> valid ; coherence (('not' ('not' p)) => q) => (p => q) is valid proof thus (('not' ('not' p)) => q) => (p => q) in TAUT A by Th28; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:54 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ( ('not' ('not' p)) => q is valid iff p => q is valid ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ( ('not' ('not' p)) => q is valid iff p => q is valid ) let p, q be Element of CQC-WFF A; ::_thesis: ( ('not' ('not' p)) => q is valid iff p => q is valid ) thus ( ('not' ('not' p)) => q is valid implies p => q is valid ) ::_thesis: ( p => q is valid implies ('not' ('not' p)) => q is valid ) proof assume A1: ('not' ('not' p)) => q is valid ; ::_thesis: p => q is valid (('not' ('not' p)) => q) => (p => q) is valid ; hence p => q is valid by A1, CQC_THE1:65; ::_thesis: verum end; assume A2: p => q is valid ; ::_thesis: ('not' ('not' p)) => q is valid ('not' ('not' p)) => p is valid ; hence ('not' ('not' p)) => q is valid by A2, Th42; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q be Element of CQC-WFF A; clusterK170(A,(p => ('not' ('not' q))),(p => q)) -> valid ; coherence (p => ('not' ('not' q))) => (p => q) is valid proof thus (p => ('not' ('not' q))) => (p => q) in TAUT A by Th29; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:55 for A being QC-alphabet for p, q being Element of CQC-WFF A holds ( p => ('not' ('not' q)) is valid iff p => q is valid ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds ( p => ('not' ('not' q)) is valid iff p => q is valid ) let p, q be Element of CQC-WFF A; ::_thesis: ( p => ('not' ('not' q)) is valid iff p => q is valid ) thus ( p => ('not' ('not' q)) is valid implies p => q is valid ) ::_thesis: ( p => q is valid implies p => ('not' ('not' q)) is valid ) proof assume A1: p => ('not' ('not' q)) is valid ; ::_thesis: p => q is valid (p => ('not' ('not' q))) => (p => q) is valid ; hence p => q is valid by A1, CQC_THE1:65; ::_thesis: verum end; assume A2: p => q is valid ; ::_thesis: p => ('not' ('not' q)) is valid q => ('not' ('not' q)) is valid ; hence p => ('not' ('not' q)) is valid by A2, Th42; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q be Element of CQC-WFF A; clusterK170(A,(p => ('not' q)),(q => ('not' p))) -> valid ; coherence (p => ('not' q)) => (q => ('not' p)) is valid proof thus (p => ('not' q)) => (q => ('not' p)) in TAUT A by Th30; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:56 for A being QC-alphabet for p, q being Element of CQC-WFF A st p => ('not' q) is valid holds q => ('not' p) is valid proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A st p => ('not' q) is valid holds q => ('not' p) is valid let p, q be Element of CQC-WFF A; ::_thesis: ( p => ('not' q) is valid implies q => ('not' p) is valid ) assume A1: p => ('not' q) is valid ; ::_thesis: q => ('not' p) is valid (p => ('not' q)) => (q => ('not' p)) is valid ; hence q => ('not' p) is valid by A1, CQC_THE1:65; ::_thesis: verum end; registration let A be QC-alphabet ; let p, q be Element of CQC-WFF A; clusterK170(A,(('not' p) => q),(('not' q) => p)) -> valid ; coherence (('not' p) => q) => (('not' q) => p) is valid proof thus (('not' p) => q) => (('not' q) => p) in TAUT A by Th31; :: according to CQC_THE1:def_10 ::_thesis: verum end; end; theorem :: LUKASI_1:57 for A being QC-alphabet for p, q being Element of CQC-WFF A st ('not' p) => q is valid holds ('not' q) => p is valid proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A st ('not' p) => q is valid holds ('not' q) => p is valid let p, q be Element of CQC-WFF A; ::_thesis: ( ('not' p) => q is valid implies ('not' q) => p is valid ) assume A1: ('not' p) => q is valid ; ::_thesis: ('not' q) => p is valid (('not' p) => q) => (('not' q) => p) is valid ; hence ('not' q) => p is valid by A1, CQC_THE1:65; ::_thesis: verum end; theorem :: LUKASI_1:58 for A being QC-alphabet for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => q holds X |- (q => r) => (p => r) proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => q holds X |- (q => r) => (p => r) let p, q, r be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => q holds X |- (q => r) => (p => r) let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => q implies X |- (q => r) => (p => r) ) assume A1: X |- p => q ; ::_thesis: X |- (q => r) => (p => r) X |- (p => q) => ((q => r) => (p => r)) by CQC_THE1:59; hence X |- (q => r) => (p => r) by A1, CQC_THE1:55; ::_thesis: verum end; theorem Th59: :: LUKASI_1:59 for A being QC-alphabet for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => q & X |- q => r holds X |- p => r proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => q & X |- q => r holds X |- p => r let p, q, r be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => q & X |- q => r holds X |- p => r let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => q & X |- q => r implies X |- p => r ) assume that A1: X |- p => q and A2: X |- q => r ; ::_thesis: X |- p => r X |- (p => q) => ((q => r) => (p => r)) by CQC_THE1:59; then X |- (q => r) => (p => r) by A1, CQC_THE1:55; hence X |- p => r by A2, CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:60 for A being QC-alphabet for p being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds X |- p => p by CQC_THE1:59; theorem :: LUKASI_1:61 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p holds X |- q => p proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p holds X |- q => p let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p holds X |- q => p let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p implies X |- q => p ) assume A1: X |- p ; ::_thesis: X |- q => p X |- p => (q => p) by CQC_THE1:59; hence X |- q => p by A1, CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:62 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p holds X |- (p => q) => q proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p holds X |- (p => q) => q let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p holds X |- (p => q) => q let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p implies X |- (p => q) => q ) assume A1: X |- p ; ::_thesis: X |- (p => q) => q X |- p => ((p => q) => q) by CQC_THE1:59; hence X |- (p => q) => q by A1, CQC_THE1:55; ::_thesis: verum end; theorem Th63: :: LUKASI_1:63 for A being QC-alphabet for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (q => r) holds X |- q => (p => r) proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (q => r) holds X |- q => (p => r) let p, q, r be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => (q => r) holds X |- q => (p => r) let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => (q => r) implies X |- q => (p => r) ) assume A1: X |- p => (q => r) ; ::_thesis: X |- q => (p => r) X |- (p => (q => r)) => (q => (p => r)) by CQC_THE1:59; hence X |- q => (p => r) by A1, CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:64 for A being QC-alphabet for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (q => r) & X |- q holds X |- p => r proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (q => r) & X |- q holds X |- p => r let p, q, r be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => (q => r) & X |- q holds X |- p => r let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => (q => r) & X |- q implies X |- p => r ) assume X |- p => (q => r) ; ::_thesis: ( not X |- q or X |- p => r ) then X |- q => (p => r) by Th63; hence ( not X |- q or X |- p => r ) by CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:65 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (p => q) holds X |- p => q proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (p => q) holds X |- p => q let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => (p => q) holds X |- p => q let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => (p => q) implies X |- p => q ) assume A1: X |- p => (p => q) ; ::_thesis: X |- p => q X |- (p => (p => q)) => (p => q) by CQC_THE1:59; hence X |- p => q by A1, CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:66 for A being QC-alphabet for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- (p => q) => r holds X |- q => r proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- (p => q) => r holds X |- q => r let p, q, r be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- (p => q) => r holds X |- q => r let X be Subset of (CQC-WFF A); ::_thesis: ( X |- (p => q) => r implies X |- q => r ) assume A1: X |- (p => q) => r ; ::_thesis: X |- q => r X |- ((p => q) => r) => (q => r) by CQC_THE1:59; hence X |- q => r by A1, CQC_THE1:55; ::_thesis: verum end; theorem Th67: :: LUKASI_1:67 for A being QC-alphabet for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (q => r) holds X |- (p => q) => (p => r) proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (q => r) holds X |- (p => q) => (p => r) let p, q, r be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => (q => r) holds X |- (p => q) => (p => r) let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => (q => r) implies X |- (p => q) => (p => r) ) assume A1: X |- p => (q => r) ; ::_thesis: X |- (p => q) => (p => r) X |- (p => (q => r)) => ((p => q) => (p => r)) by CQC_THE1:59; hence X |- (p => q) => (p => r) by A1, CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:68 for A being QC-alphabet for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (q => r) & X |- p => q holds X |- p => r proof let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => (q => r) & X |- p => q holds X |- p => r let p, q, r be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => (q => r) & X |- p => q holds X |- p => r let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => (q => r) & X |- p => q implies X |- p => r ) assume X |- p => (q => r) ; ::_thesis: ( not X |- p => q or X |- p => r ) then X |- (p => q) => (p => r) by Th67; hence ( not X |- p => q or X |- p => r ) by CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:69 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds ( X |- ('not' p) => ('not' q) iff X |- q => p ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds ( X |- ('not' p) => ('not' q) iff X |- q => p ) let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) holds ( X |- ('not' p) => ('not' q) iff X |- q => p ) let X be Subset of (CQC-WFF A); ::_thesis: ( X |- ('not' p) => ('not' q) iff X |- q => p ) thus ( X |- ('not' p) => ('not' q) implies X |- q => p ) ::_thesis: ( X |- q => p implies X |- ('not' p) => ('not' q) ) proof assume A1: X |- ('not' p) => ('not' q) ; ::_thesis: X |- q => p X |- (('not' p) => ('not' q)) => (q => p) by CQC_THE1:59; hence X |- q => p by A1, CQC_THE1:55; ::_thesis: verum end; assume A2: X |- q => p ; ::_thesis: X |- ('not' p) => ('not' q) X |- (q => p) => (('not' p) => ('not' q)) by CQC_THE1:59; hence X |- ('not' p) => ('not' q) by A2, CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:70 for A being QC-alphabet for p being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds ( X |- 'not' ('not' p) iff X |- p ) proof let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds ( X |- 'not' ('not' p) iff X |- p ) let p be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) holds ( X |- 'not' ('not' p) iff X |- p ) let X be Subset of (CQC-WFF A); ::_thesis: ( X |- 'not' ('not' p) iff X |- p ) thus ( X |- 'not' ('not' p) implies X |- p ) ::_thesis: ( X |- p implies X |- 'not' ('not' p) ) proof assume A1: X |- 'not' ('not' p) ; ::_thesis: X |- p X |- ('not' ('not' p)) => p by CQC_THE1:59; hence X |- p by A1, CQC_THE1:55; ::_thesis: verum end; assume A2: X |- p ; ::_thesis: X |- 'not' ('not' p) X |- p => ('not' ('not' p)) by CQC_THE1:59; hence X |- 'not' ('not' p) by A2, CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:71 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds ( X |- p => ('not' ('not' q)) iff X |- p => q ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds ( X |- p => ('not' ('not' q)) iff X |- p => q ) let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) holds ( X |- p => ('not' ('not' q)) iff X |- p => q ) let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => ('not' ('not' q)) iff X |- p => q ) thus ( X |- p => ('not' ('not' q)) implies X |- p => q ) ::_thesis: ( X |- p => q implies X |- p => ('not' ('not' q)) ) proof assume A1: X |- p => ('not' ('not' q)) ; ::_thesis: X |- p => q X |- (p => ('not' ('not' q))) => (p => q) by CQC_THE1:59; hence X |- p => q by A1, CQC_THE1:55; ::_thesis: verum end; assume A2: X |- p => q ; ::_thesis: X |- p => ('not' ('not' q)) X |- q => ('not' ('not' q)) by CQC_THE1:59; hence X |- p => ('not' ('not' q)) by A2, Th59; ::_thesis: verum end; theorem :: LUKASI_1:72 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds ( X |- ('not' ('not' p)) => q iff X |- p => q ) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) holds ( X |- ('not' ('not' p)) => q iff X |- p => q ) let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) holds ( X |- ('not' ('not' p)) => q iff X |- p => q ) let X be Subset of (CQC-WFF A); ::_thesis: ( X |- ('not' ('not' p)) => q iff X |- p => q ) thus ( X |- ('not' ('not' p)) => q implies X |- p => q ) ::_thesis: ( X |- p => q implies X |- ('not' ('not' p)) => q ) proof assume A1: X |- ('not' ('not' p)) => q ; ::_thesis: X |- p => q X |- (('not' ('not' p)) => q) => (p => q) by CQC_THE1:59; hence X |- p => q by A1, CQC_THE1:55; ::_thesis: verum end; assume A2: X |- p => q ; ::_thesis: X |- ('not' ('not' p)) => q X |- ('not' ('not' p)) => p by CQC_THE1:59; hence X |- ('not' ('not' p)) => q by A2, Th59; ::_thesis: verum end; theorem Th73: :: LUKASI_1:73 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => ('not' q) holds X |- q => ('not' p) proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => ('not' q) holds X |- q => ('not' p) let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => ('not' q) holds X |- q => ('not' p) let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => ('not' q) implies X |- q => ('not' p) ) assume A1: X |- p => ('not' q) ; ::_thesis: X |- q => ('not' p) X |- (p => ('not' q)) => (q => ('not' p)) by CQC_THE1:59; hence X |- q => ('not' p) by A1, CQC_THE1:55; ::_thesis: verum end; theorem Th74: :: LUKASI_1:74 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- ('not' p) => q holds X |- ('not' q) => p proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- ('not' p) => q holds X |- ('not' q) => p let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- ('not' p) => q holds X |- ('not' q) => p let X be Subset of (CQC-WFF A); ::_thesis: ( X |- ('not' p) => q implies X |- ('not' q) => p ) assume A1: X |- ('not' p) => q ; ::_thesis: X |- ('not' q) => p X |- (('not' p) => q) => (('not' q) => p) by CQC_THE1:59; hence X |- ('not' q) => p by A1, CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:75 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => ('not' q) & X |- q holds X |- 'not' p proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- p => ('not' q) & X |- q holds X |- 'not' p let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- p => ('not' q) & X |- q holds X |- 'not' p let X be Subset of (CQC-WFF A); ::_thesis: ( X |- p => ('not' q) & X |- q implies X |- 'not' p ) assume X |- p => ('not' q) ; ::_thesis: ( not X |- q or X |- 'not' p ) then X |- q => ('not' p) by Th73; hence ( not X |- q or X |- 'not' p ) by CQC_THE1:55; ::_thesis: verum end; theorem :: LUKASI_1:76 for A being QC-alphabet for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- ('not' p) => q & X |- 'not' q holds X |- p proof let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A for X being Subset of (CQC-WFF A) st X |- ('not' p) => q & X |- 'not' q holds X |- p let p, q be Element of CQC-WFF A; ::_thesis: for X being Subset of (CQC-WFF A) st X |- ('not' p) => q & X |- 'not' q holds X |- p let X be Subset of (CQC-WFF A); ::_thesis: ( X |- ('not' p) => q & X |- 'not' q implies X |- p ) assume X |- ('not' p) => q ; ::_thesis: ( not X |- 'not' q or X |- p ) then X |- ('not' q) => p by Th74; hence ( not X |- 'not' q or X |- p ) by CQC_THE1:55; ::_thesis: verum end;