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

Theoremlooinvdc 901 The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 881, we can see that this expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability condition). (Contributed by NM, 12-Aug-2004.)
DECID

1.2.10  Miscellaneous theorems of propositional calculus

Theorempm5.21nd 902 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)

Theorempm5.35 903 Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theorempm5.54dc 904 A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
DECID

Theorembaib 905 Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)

Theorembaibr 906 Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)

Theoremrbaib 907 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremrbaibr 908 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theorembaibd 909 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremrbaibd 910 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theorempm5.44 911 Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theorempm5.6dc 912 Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 913). (Contributed by Jim Kingdon, 2-Apr-2018.)
DECID

Theorempm5.6r 913 Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If is decidable, the converse also holds (see pm5.6dc 912). (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremorcanai 914 Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)

Theoremintnan 915 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)

Theoremintnanr 916 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)

Theoremintnand 917 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)

Theoremintnanrd 918 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)

Theoremdcan 919 A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
DECID DECID DECID

Theoremdcor 920 A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
DECID DECID DECID

Theoremdcbi 921 An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
DECID DECID DECID

Theoremannimdc 922 Express conjunction in terms of implication. The forward direction, annimim 676, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)
DECID DECID

Theorempm4.55dc 923 Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
DECID DECID

Theoremorandc 924 Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)
DECID DECID

Theoremmpbiran 925 Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)

Theoremmpbiran2 926 Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)

Theoremmpbir2an 927 Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.) (Revised by NM, 9-Jan-2015.)

Theoremmpbi2and 928 Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Theoremmpbir2and 929 Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Theorempm5.62dc 930 Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
DECID

Theorempm5.63dc 931 Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
DECID

Theorembianfi 932 A wff conjoined with falsehood is false. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)

Theorembianfd 933 A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.)

Theorempm4.43 934 Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)

Theorempm4.82 935 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Theorempm4.83dc 936 Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 851, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
DECID

Theorembiantr 937 A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)

Theoremorbididc 938 Disjunction distributes over the biconditional, for a decidable proposition. Based on an axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by Jim Kingdon, 2-Apr-2018.)
DECID

Theorempm5.7dc 939 Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 938. (Contributed by Jim Kingdon, 2-Apr-2018.)
DECID

Theorembigolden 940 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)

Theoremanordc 941 Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 744, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)
DECID DECID

Theorempm3.11dc 942 Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 744, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
DECID DECID

Theorempm3.12dc 943 Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
DECID DECID

Theorempm3.13dc 944 Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 743, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
DECID DECID

Theoremdn1dc 945 DN1 for decidable propositions. Without the decidability conditions, DN1 can serve as a single axiom for Boolean algebra. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jim Kingdon, 22-Apr-2018.)
DECID DECID DECID DECID

Theorempm5.71dc 946 Decidable proposition version of theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for decidability by Jim Kingdon, 19-Apr-2018.)
DECID

Theorempm5.75 947 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 948 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremccase 949 Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)

Theoremccased 950 Deduction for combining cases. (Contributed by NM, 9-May-2004.)

Theoremccase2 951 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)

Theoremniabn 952 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)

Theoremdedlem0a 953 Alternate version of dedlema 954. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)

Theoremdedlema 954 Lemma for iftrue 3510. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremdedlemb 955 Lemma for iffalse 3513. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theorempm4.42r 956 One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)

Theoremninba 957 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)

Theoremprlem1 958 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 959 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 960 A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

Theoremrnlem 961 Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

1.2.11  Abbreviated conjunction and disjunction of three wff's

Syntaxw3o 962 Extend wff definition to include 3-way disjunction ('or').

Syntaxw3a 963 Extend wff definition to include 3-way conjunction ('and').

Definitiondf-3or 964 Define disjunction ('or') of 3 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 757. (Contributed by NM, 8-Apr-1994.)

Definitiondf-3an 965 Define conjunction ('and') of 3 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 399. (Contributed by NM, 8-Apr-1994.)

Theorem3orass 966 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)

Theorem3anass 967 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)

Theorem3anrot 968 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)

Theorem3orrot 969 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)

Theorem3ancoma 970 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3ancomb 971 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3orcomb 972 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)

Theorem3anrev 973 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3anan32 974 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theorem3anan12 975 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremanandi3 976 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)

Theoremanandi3r 977 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)

Theorem3ioran 978 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)

Theorem3simpa 979 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3simpb 980 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theorem3simpc 981 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremsimp1 982 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theoremsimp2 983 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theoremsimp3 984 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Theoremsimpl1 985 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpl2 986 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpl3 987 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpr1 988 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpr2 989 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimpr3 990 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremsimp1i 991 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)

Theoremsimp2i 992 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)

Theoremsimp3i 993 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)

Theoremsimp1d 994 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)

Theoremsimp2d 995 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)

Theoremsimp3d 996 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)

Theoremsimp1bi 997 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsimp2bi 998 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsimp3bi 999 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theorem3adant1 1000 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)

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