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Theorem List for Intuitionistic Logic Explorer - 901-1000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3simpb 901 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((φ ψ χ) → (φ χ))
 
Theorem3simpc 902 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((φ ψ χ) → (ψ χ))
 
Theoremsimp1 903 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((φ ψ χ) → φ)
 
Theoremsimp2 904 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((φ ψ χ) → ψ)
 
Theoremsimp3 905 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((φ ψ χ) → χ)
 
Theoremsimpl1 906 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((φ ψ χ) θ) → φ)
 
Theoremsimpl2 907 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((φ ψ χ) θ) → ψ)
 
Theoremsimpl3 908 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((φ ψ χ) θ) → χ)
 
Theoremsimpr1 909 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((φ (ψ χ θ)) → ψ)
 
Theoremsimpr2 910 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((φ (ψ χ θ)) → χ)
 
Theoremsimpr3 911 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((φ (ψ χ θ)) → θ)
 
Theoremsimp1i 912 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(φ ψ χ)       φ
 
Theoremsimp2i 913 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(φ ψ χ)       ψ
 
Theoremsimp3i 914 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(φ ψ χ)       χ
 
Theoremsimp1d 915 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(φ → (ψ χ θ))       (φψ)
 
Theoremsimp2d 916 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(φ → (ψ χ θ))       (φχ)
 
Theoremsimp3d 917 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(φ → (ψ χ θ))       (φθ)
 
Theoremsimp1bi 918 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φ ↔ (ψ χ θ))       (φψ)
 
Theoremsimp2bi 919 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φ ↔ (ψ χ θ))       (φχ)
 
Theoremsimp3bi 920 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φ ↔ (ψ χ θ))       (φθ)
 
Theorem3adant1 921 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((φ ψ) → χ)       ((θ φ ψ) → χ)
 
Theorem3adant2 922 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((φ ψ) → χ)       ((φ θ ψ) → χ)
 
Theorem3adant3 923 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((φ ψ) → χ)       ((φ ψ θ) → χ)
 
Theorem3ad2ant1 924 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(φχ)       ((φ ψ θ) → χ)
 
Theorem3ad2ant2 925 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(φχ)       ((ψ φ θ) → χ)
 
Theorem3ad2ant3 926 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(φχ)       ((ψ θ φ) → χ)
 
Theoremsimp1l 927 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((φ ψ) χ θ) → φ)
 
Theoremsimp1r 928 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((φ ψ) χ θ) → ψ)
 
Theoremsimp2l 929 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((φ (ψ χ) θ) → ψ)
 
Theoremsimp2r 930 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((φ (ψ χ) θ) → χ)
 
Theoremsimp3l 931 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((φ ψ (χ θ)) → χ)
 
Theoremsimp3r 932 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((φ ψ (χ θ)) → θ)
 
Theoremsimp11 933 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((φ ψ χ) θ τ) → φ)
 
Theoremsimp12 934 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((φ ψ χ) θ τ) → ψ)
 
Theoremsimp13 935 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((φ ψ χ) θ τ) → χ)
 
Theoremsimp21 936 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((φ (ψ χ θ) τ) → ψ)
 
Theoremsimp22 937 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((φ (ψ χ θ) τ) → χ)
 
Theoremsimp23 938 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((φ (ψ χ θ) τ) → θ)
 
Theoremsimp31 939 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((φ ψ (χ θ τ)) → χ)
 
Theoremsimp32 940 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((φ ψ (χ θ τ)) → θ)
 
Theoremsimp33 941 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((φ ψ (χ θ τ)) → τ)
 
Theoremsimpll1 942 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ) τ) → φ)
 
Theoremsimpll2 943 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ) τ) → ψ)
 
Theoremsimpll3 944 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ) τ) → χ)
 
Theoremsimplr1 945 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ)) τ) → φ)
 
Theoremsimplr2 946 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ)) τ) → ψ)
 
Theoremsimplr3 947 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ)) τ) → χ)
 
Theoremsimprl1 948 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ χ) θ)) → φ)
 
Theoremsimprl2 949 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ χ) θ)) → ψ)
 
Theoremsimprl3 950 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ χ) θ)) → χ)
 
Theoremsimprr1 951 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (θ (φ ψ χ))) → φ)
 
Theoremsimprr2 952 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (θ (φ ψ χ))) → ψ)
 
Theoremsimprr3 953 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (θ (φ ψ χ))) → χ)
 
Theoremsimpl1l 954 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ) χ θ) τ) → φ)
 
Theoremsimpl1r 955 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ) χ θ) τ) → ψ)
 
Theoremsimpl2l 956 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ (φ ψ) θ) τ) → φ)
 
Theoremsimpl2r 957 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ (φ ψ) θ) τ) → ψ)
 
Theoremsimpl3l 958 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ θ (φ ψ)) τ) → φ)
 
Theoremsimpl3r 959 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ θ (φ ψ)) τ) → ψ)
 
Theoremsimpr1l 960 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ) χ θ)) → φ)
 
Theoremsimpr1r 961 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ) χ θ)) → ψ)
 
Theoremsimpr2l 962 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ (φ ψ) θ)) → φ)
 
Theoremsimpr2r 963 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ (φ ψ) θ)) → ψ)
 
Theoremsimpr3l 964 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ θ (φ ψ))) → φ)
 
Theoremsimpr3r 965 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ θ (φ ψ))) → ψ)
 
Theoremsimp1ll 966 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ) χ) θ τ) → φ)
 
Theoremsimp1lr 967 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ) χ) θ τ) → ψ)
 
Theoremsimp1rl 968 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ (φ ψ)) θ τ) → φ)
 
Theoremsimp1rr 969 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ (φ ψ)) θ τ) → ψ)
 
Theoremsimp2ll 970 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((θ ((φ ψ) χ) τ) → φ)
 
Theoremsimp2lr 971 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((θ ((φ ψ) χ) τ) → ψ)
 
Theoremsimp2rl 972 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((θ (χ (φ ψ)) τ) → φ)
 
Theoremsimp2rr 973 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((θ (χ (φ ψ)) τ) → ψ)
 
Theoremsimp3ll 974 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((θ τ ((φ ψ) χ)) → φ)
 
Theoremsimp3lr 975 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((θ τ ((φ ψ) χ)) → ψ)
 
Theoremsimp3rl 976 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((θ τ (χ (φ ψ))) → φ)
 
Theoremsimp3rr 977 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((θ τ (χ (φ ψ))) → ψ)
 
Theoremsimpl11 978 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η) → φ)
 
Theoremsimpl12 979 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η) → ψ)
 
Theoremsimpl13 980 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η) → χ)
 
Theoremsimpl21 981 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η) → φ)
 
Theoremsimpl22 982 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η) → ψ)
 
Theoremsimpl23 983 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η) → χ)
 
Theoremsimpl31 984 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η) → φ)
 
Theoremsimpl32 985 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η) → ψ)
 
Theoremsimpl33 986 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η) → χ)
 
Theoremsimpr11 987 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ)) → φ)
 
Theoremsimpr12 988 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ)) → ψ)
 
Theoremsimpr13 989 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ)) → χ)
 
Theoremsimpr21 990 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ)) → φ)
 
Theoremsimpr22 991 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ)) → ψ)
 
Theoremsimpr23 992 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ)) → χ)
 
Theoremsimpr31 993 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ))) → φ)
 
Theoremsimpr32 994 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ))) → ψ)
 
Theoremsimpr33 995 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ))) → χ)
 
Theoremsimp1l1 996 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ) τ η) → φ)
 
Theoremsimp1l2 997 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ) τ η) → ψ)
 
Theoremsimp1l3 998 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ) τ η) → χ)
 
Theoremsimp1r1 999 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ)) τ η) → φ)
 
Theoremsimp1r2 1000 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ)) τ η) → ψ)
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