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Theorem List for Metamath Proof Explorer - 901-1000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm5.7 901 Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 899. (Contributed by Roy F. Longton, 21-Jun-2005.)

Theorembigolden 902 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)

Theorempm5.71 903 Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)

Theorempm5.75 904 Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.)

Theorembimsc1 905 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)

Theorem4exmid 906 The disjunction of the four possible combinations of two wffs and their negations is always true. (Contributed by David Abernethy, 28-Jan-2014.)

Theoremecase2d 907 Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.)

Theoremecase3 908 Inference for elimination by cases. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)

Theoremecase 909 Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)

Theoremecase3d 910 Deduction for elimination by cases. (Contributed by NM, 2-May-1996.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremecased 911 Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.)

Theoremecase3ad 912 Deduction for elimination by cases. (Contributed by NM, 24-May-2013.)

Theoremccase 913 Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)

Theoremccased 914 Deduction for combining cases. (Contributed by NM, 9-May-2004.)

Theoremccase2 915 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)

Theorem4cases 916 Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.)

Theorem4casesdan 917 Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.)

Theoremniabn 918 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)

Theoremdedlem0a 919 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)

Theoremdedlem0b 920 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)

Theoremdedlema 921 Lemma for weak deduction theorem. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremdedlemb 922 Lemma for weak deduction theorem. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremelimh 923 Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page. (Contributed by NM, 26-Jun-2002.)

Theoremdedt 924 The weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page. (Contributed by NM, 26-Jun-2002.)

Theoremcon3th 925 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 128 demonstrates the use of the weak deduction theorem dedt 924 to derive it from con3i 129. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.)

Theoremconsensus 926 The consensus theorem. This theorem and its dual (with and interchanged) are commonly used in computer logic design to eliminate redundant terms from Boolean expressions. Specifically, we prove that the term on the left-hand side is redundant. (Contributed by NM, 16-May-2003.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)

Theorempm4.42 927 Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Roy F. Longton, 21-Jun-2005.)

Theoremninba 928 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)

Theoremprlem1 929 A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)

Theoremprlem2 930 A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)

Theoremoplem1 931 A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.)

Theoremrnlem 932 Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdn1 933 A single axiom for Boolean algebra known as DN1. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jeffrey Hankins, 3-Jul-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)

Theoremjaoi2 934 Inference removing a negated conjunct in a disjunction of an antecedent if this conjunct is part of the disjunction. (Contributed by Alexander van der Vekens, 3-Nov-2017.)

1.2.8  Abbreviated conjunction and disjunction of three wff's

Syntaxw3o 935 Extend wff definition to include 3-way disjunction ('or').

Syntaxw3a 936 Extend wff definition to include 3-way conjunction ('and').

Definitiondf-3or 937 Define disjunction ('or') of three wff's. Definition *2.33 of [WhiteheadRussell] p. 105. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law orass 511. (Contributed by NM, 8-Apr-1994.)

Definitiondf-3an 938 Define conjunction ('and') of three wff's. Definition *4.34 of [WhiteheadRussell] p. 118. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law anass 631. (Contributed by NM, 8-Apr-1994.)

Theorem3orass 939 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)

Theorem3anass 940 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)

Theorem3anrot 941 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)

Theorem3orrot 942 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)

Theorem3ancoma 943 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3orcoma 944 Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theorem3ancomb 945 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3orcomb 946 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)

Theorem3anrev 947 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3anan32 948 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theorem3anan12 949 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theorem3anor 950 Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.)

Theorem3ianor 951 Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theorem3ioran 952 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)

Theorem3oran 953 Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.)

Theorem3simpa 954 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3simpb 955 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3simpc 956 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremsimp1 957 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theoremsimp2 958 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theoremsimp3 959 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theoremsimpl1 960 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpl2 961 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpl3 962 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpr1 963 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpr2 964 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpr3 965 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimp1i 966 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)

Theoremsimp2i 967 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)

Theoremsimp3i 968 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)

Theoremsimp1d 969 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)

Theoremsimp2d 970 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)

Theoremsimp3d 971 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)

Theoremsimp1bi 972 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsimp2bi 973 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsimp3bi 974 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theorem3adant1 975 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)

Theorem3adant2 976 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)

Theorem3adant3 977 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)

Theorem3ad2ant1 978 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)

Theorem3ad2ant2 979 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)

Theorem3ad2ant3 980 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)

Theoremsimp1l 981 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)

Theoremsimp1r 982 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)

Theoremsimp2l 983 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)

Theoremsimp2r 984 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)

Theoremsimp3l 985 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)

Theoremsimp3r 986 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)

Theoremsimp11 987 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimp12 988 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimp13 989 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimp21 990 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimp22 991 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimp23 992 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimp31 993 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimp32 994 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimp33 995 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)

Theoremsimpll1 996 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)

Theoremsimpll2 997 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)

Theoremsimpll3 998 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)

Theoremsimplr1 999 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)

Theoremsimplr2 1000 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)

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