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Type | Label | Description |
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Statement | ||
Theorem | pm2.54dc 801 | Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 651, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.) |
DECID | ||
Theorem | dfordc 802 | Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 651, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.) |
DECID | ||
Theorem | pm2.25dc 803 | Elimination of disjunction based on a disjunction, for a decidable proposition. Based on theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
DECID | ||
Theorem | pm2.68dc 804 | Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 677 and one half of dfor2dc 805. (Contributed by Jim Kingdon, 27-Mar-2018.) |
DECID | ||
Theorem | dfor2dc 805 | Logical 'or' expressed in terms of implication only, for a decidable proposition. Based on theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 27-Mar-2018.) |
DECID | ||
Theorem | imimorbdc 806 | Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.) |
DECID | ||
Theorem | imordc 807 | Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 808, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
DECID | ||
Theorem | imorr 808 | Implication in terms of disjunction. One direction of theorem *4.6 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as seen at imordc 807. (Contributed by Jim Kingdon, 21-Jul-2018.) |
Theorem | pm4.62dc 809 | Implication in terms of disjunction. Like Theorem *4.62 of [WhiteheadRussell] p. 120, but for a decidable antecedent. (Contributed by Jim Kingdon, 21-Apr-2018.) |
DECID | ||
Theorem | ianordc 810 | Negated conjunction in terms of disjunction (DeMorgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120, but where one proposition is decidable. The reverse direction, pm3.14 680, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.) |
DECID | ||
Theorem | oibabs 811 | Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) |
Theorem | pm4.64dc 812 | Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 651, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.) |
DECID | ||
Theorem | pm4.66dc 813 | Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability condition. (Contributed by Jim Kingdon, 2-May-2018.) |
DECID | ||
Theorem | pm4.52im 814 | One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.) |
Theorem | pm4.53r 815 | One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.) |
Theorem | pm4.54dc 816 | Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.) |
DECID DECID | ||
Theorem | pm4.56 817 | Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
Theorem | oranim 818 | Disjunction in terms of conjunction (DeMorgan's law). One direction of Theorem *4.57 of [WhiteheadRussell] p. 120. The converse does not hold intuitionistically but does hold in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.) |
Theorem | pm4.78i 819 | Implication distributes over disjunction. One direction of Theorem *4.78 of [WhiteheadRussell] p. 121. The converse holds in classical logic. (Contributed by Jim Kingdon, 15-Jan-2018.) |
Theorem | pm4.79dc 820 | Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.) |
DECID DECID | ||
Theorem | pm5.17dc 821 | Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.) |
DECID | ||
Theorem | pm2.85dc 822 | Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.) |
DECID | ||
Theorem | orimdidc 823 | Disjunction distributes over implication. The forward direction, pm2.76 732, is valid intuitionistically. The reverse direction holds if is decidable, as can be seen at pm2.85dc 822. (Contributed by Jim Kingdon, 1-Apr-2018.) |
DECID | ||
Theorem | pm2.26dc 824 | Decidable proposition version of theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.) |
DECID | ||
Theorem | pm4.81dc 825 | Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 633 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.) |
DECID | ||
Theorem | pm5.11dc 826 | A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.) |
DECID DECID | ||
Theorem | pm5.12dc 827 | Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.) |
DECID | ||
Theorem | pm5.14dc 828 | A decidable proposition is implied by or implies other propositions. Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.) |
DECID | ||
Theorem | pm5.13dc 829 | An implication holds in at least one direction, where one proposition is decidable. Based on theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.) |
DECID | ||
Theorem | pm5.55dc 830 | A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.) |
DECID | ||
Theorem | peircedc 831 | Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 755, condc 760, or notnotrdc 762 in the sense of expressing the "difference" between an intuitionistic system of propositional calculus and a classical system. In intuitionistic logic, it only holds for decidable propositions. (Contributed by Jim Kingdon, 3-Jul-2018.) |
DECID | ||
Theorem | looinvdc 832 | The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 805, 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 | ||
Theorem | dftest 833 |
A proposition is testable iff its negative or double-negative is true.
See Chapter 2 [Moschovakis] p. 2.
Our notation for testability is DECID before the formula in question. For example, DECID corresponds to "x = y is testable". (Contributed by David A. Wheeler, 13-Aug-2018.) |
DECID | ||
Theorem | testbitestn 834 | A proposition is testable iff its negation is testable. See also dcn 757 (which could be read as "Decidability implies testability"). (Contributed by David A. Wheeler, 6-Dec-2018.) |
DECID DECID | ||
Theorem | stabtestimpdc 835 | "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.) |
STAB DECID DECID | ||
Theorem | pm5.21nd 836 | Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.) |
Theorem | pm5.35 837 | Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm5.54dc 838 | 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 | ||
Theorem | baib 839 | Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.) |
Theorem | baibr 840 | Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.) |
Theorem | rbaib 841 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Theorem | rbaibr 842 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Theorem | baibd 843 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Theorem | rbaibd 844 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Theorem | pm5.44 845 | Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm5.6dc 846 | 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 847). (Contributed by Jim Kingdon, 2-Apr-2018.) |
DECID | ||
Theorem | pm5.6r 847 | 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 846). (Contributed by Jim Kingdon, 4-Aug-2018.) |
Theorem | orcanai 848 | Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.) |
Theorem | intnan 849 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.) |
Theorem | intnanr 850 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.) |
Theorem | intnand 851 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.) |
Theorem | intnanrd 852 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.) |
Theorem | dcan 853 | A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) |
DECID DECID DECID | ||
Theorem | dcor 854 | A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.) |
DECID DECID DECID | ||
Theorem | dcbi 855 | An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) |
DECID DECID DECID | ||
Theorem | annimdc 856 | Express conjunction in terms of implication. The forward direction, annimim 793, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.) |
DECID DECID | ||
Theorem | pm4.55dc 857 | Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.) |
DECID DECID | ||
Theorem | mpbiran 858 | Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.) |
Theorem | mpbiran2 859 | Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.) |
Theorem | mpbir2an 860 | Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.) (Revised by NM, 9-Jan-2015.) |
Theorem | mpbi2and 861 | Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) |
Theorem | mpbir2and 862 | Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) |
Theorem | pm5.62dc 863 | Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.) |
DECID | ||
Theorem | pm5.63dc 864 | Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.) |
DECID | ||
Theorem | bianfi 865 | A wff conjoined with falsehood is false. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) |
Theorem | bianfd 866 | A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.) |
Theorem | pm4.43 867 | Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) |
Theorem | pm4.82 868 | Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm4.83dc 869 | Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 773, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.) |
DECID | ||
Theorem | biantr 870 | A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.) |
Theorem | orbididc 871 | 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 | ||
Theorem | pm5.7dc 872 | Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 871. (Contributed by Jim Kingdon, 2-Apr-2018.) |
DECID | ||
Theorem | bigolden 873 | Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.) |
Theorem | anordc 874 | Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 681, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.) |
DECID DECID | ||
Theorem | pm3.11dc 875 | Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 681, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.) |
DECID DECID | ||
Theorem | pm3.12dc 876 | Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.) |
DECID DECID | ||
Theorem | pm3.13dc 877 | Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 680, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.) |
DECID DECID | ||
Theorem | dn1dc 878 | DN_{1} for decidable propositions. Without the decidability conditions, DN_{1} 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 | ||
Theorem | pm5.71dc 879 | 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 | ||
Theorem | pm5.75 880 | 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.) |
Theorem | bimsc1 881 | Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | ccase 882 | Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.) |
Theorem | ccased 883 | Deduction for combining cases. (Contributed by NM, 9-May-2004.) |
Theorem | ccase2 884 | Inference for combining cases. (Contributed by NM, 29-Jul-1999.) |
Theorem | niabn 885 | Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.) |
Theorem | dedlem0a 886 | Alternate version of dedlema 887. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) |
Theorem | dedlema 887 | Lemma for iftrue 3364. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Theorem | dedlemb 888 | Lemma for iffalse 3367. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Theorem | pm4.42r 889 | One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Theorem | ninba 890 | Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.) |
Theorem | prlem1 891 | 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.) |
Theorem | prlem2 892 | 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.) |
Theorem | oplem1 893 | 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.) |
Theorem | rnlem 894 | Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Syntax | w3o 895 | Extend wff definition to include 3-way disjunction ('or'). |
Syntax | w3a 896 | Extend wff definition to include 3-way conjunction ('and'). |
Definition | df-3or 897 | 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 694. (Contributed by NM, 8-Apr-1994.) |
Definition | df-3an 898 | 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 387. (Contributed by NM, 8-Apr-1994.) |
Theorem | 3orass 899 | Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.) |
Theorem | 3anass 900 | Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.) |
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