HomeHome Intuitionistic Logic Explorer
Theorem List (p. 10 of 102)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 901-1000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3simpa 901 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → (𝜑𝜓))
 
Theorem3simpb 902 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → (𝜑𝜒))
 
Theorem3simpc 903 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((𝜑𝜓𝜒) → (𝜓𝜒))
 
Theoremsimp1 904 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜑)
 
Theoremsimp2 905 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜓)
 
Theoremsimp3 906 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜒)
 
Theoremsimpl1 907 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜑)
 
Theoremsimpl2 908 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimpl3 909 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜒)
 
Theoremsimpr1 910 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜓)
 
Theoremsimpr2 911 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜒)
 
Theoremsimpr3 912 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜃)
 
Theoremsimp1i 913 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜑
 
Theoremsimp2i 914 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜓
 
Theoremsimp3i 915 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜒
 
Theoremsimp1d 916 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theoremsimp2d 917 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜒)
 
Theoremsimp3d 918 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜃)
 
Theoremsimp1bi 919 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theoremsimp2bi 920 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜒)
 
Theoremsimp3bi 921 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜃)
 
Theorem3adant1 922 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜃𝜑𝜓) → 𝜒)
 
Theorem3adant2 923 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜃𝜓) → 𝜒)
 
Theorem3adant3 924 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3ad2ant1 925 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3ad2ant2 926 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜑𝜃) → 𝜒)
 
Theorem3ad2ant3 927 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜃𝜑) → 𝜒)
 
Theoremsimp1l 928 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜑)
 
Theoremsimp1r 929 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜓)
 
Theoremsimp2l 930 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimp2r 931 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜒)
 
Theoremsimp3l 932 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜒)
 
Theoremsimp3r 933 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜃)
 
Theoremsimp11 934 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜑)
 
Theoremsimp12 935 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜓)
 
Theoremsimp13 936 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜒)
 
Theoremsimp21 937 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimp22 938 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜒)
 
Theoremsimp23 939 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜃)
 
Theoremsimp31 940 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜒)
 
Theoremsimp32 941 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜃)
 
Theoremsimp33 942 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜏)
 
Theoremsimpll1 943 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)
 
Theoremsimpll2 944 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimpll3 945 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜒)
 
Theoremsimplr1 946 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜑)
 
Theoremsimplr2 947 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜓)
 
Theoremsimplr3 948 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜒)
 
Theoremsimprl1 949 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜑)
 
Theoremsimprl2 950 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜓)
 
Theoremsimprl3 951 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜒)
 
Theoremsimprr1 952 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜑)
 
Theoremsimprr2 953 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜓)
 
Theoremsimprr3 954 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜒)
 
Theoremsimpl1l 955 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏) → 𝜑)
 
Theoremsimpl1r 956 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimpl2l 957 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜑)
 
Theoremsimpl2r 958 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimpl3l 959 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜑)
 
Theoremsimpl3r 960 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜓)
 
Theoremsimpr1l 961 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜑)
 
Theoremsimpr1r 962 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜓)
 
Theoremsimpr2l 963 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜑)
 
Theoremsimpr2r 964 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜓)
 
Theoremsimpr3l 965 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒𝜃 ∧ (𝜑𝜓))) → 𝜑)
 
Theoremsimpr3r 966 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒𝜃 ∧ (𝜑𝜓))) → 𝜓)
 
Theoremsimp1ll 967 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃𝜏) → 𝜑)
 
Theoremsimp1lr 968 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃𝜏) → 𝜓)
 
Theoremsimp1rl 969 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓)) ∧ 𝜃𝜏) → 𝜑)
 
Theoremsimp1rr 970 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓)) ∧ 𝜃𝜏) → 𝜓)
 
Theoremsimp2ll 971 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃 ∧ ((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜑)
 
Theoremsimp2lr 972 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃 ∧ ((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜓)
 
Theoremsimp2rl 973 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃 ∧ (𝜒 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜑)
 
Theoremsimp2rr 974 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃 ∧ (𝜒 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜓)
 
Theoremsimp3ll 975 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃𝜏 ∧ ((𝜑𝜓) ∧ 𝜒)) → 𝜑)
 
Theoremsimp3lr 976 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃𝜏 ∧ ((𝜑𝜓) ∧ 𝜒)) → 𝜓)
 
Theoremsimp3rl 977 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃𝜏 ∧ (𝜒 ∧ (𝜑𝜓))) → 𝜑)
 
Theoremsimp3rr 978 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜃𝜏 ∧ (𝜒 ∧ (𝜑𝜓))) → 𝜓)
 
Theoremsimpl11 979 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂) → 𝜑)
 
Theoremsimpl12 980 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂) → 𝜓)
 
Theoremsimpl13 981 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜂) → 𝜒)
 
Theoremsimpl21 982 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜑)
 
Theoremsimpl22 983 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremsimpl23 984 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜂) → 𝜒)
 
Theoremsimpl31 985 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜑)
 
Theoremsimpl32 986 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜓)
 
Theoremsimpl33 987 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂) → 𝜒)
 
Theoremsimpr11 988 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜑)
 
Theoremsimpr12 989 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜓)
 
Theoremsimpr13 990 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜒)
 
Theoremsimpr21 991 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜑)
 
Theoremsimpr22 992 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜓)
 
Theoremsimpr23 993 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜒)
 
Theoremsimpr31 994 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜑)
 
Theoremsimpr32 995 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜓)
 
Theoremsimpr33 996 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜒)
 
Theoremsimp1l1 997 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜑)
 
Theoremsimp1l2 998 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜓)
 
Theoremsimp1l3 999 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏𝜂) → 𝜒)
 
Theoremsimp1r1 1000 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏𝜂) → 𝜑)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10124
  Copyright terms: Public domain < Previous  Next >