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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)

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