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Theorem List for Intuitionistic Logic Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm4.66dc 801 Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability condition. (Contributed by Jim Kingdon, 2-May-2018.)
(DECID φ → ((¬ φ → ¬ ψ) ↔ (φ ¬ ψ)))
 
Theorempm4.52im 802 One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
((φ ¬ ψ) → ¬ (¬ φ ψ))
 
Theorempm4.53r 803 One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
((¬ φ ψ) → ¬ (φ ¬ ψ))
 
Theorempm4.54dc 804 Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)
(DECID φ → (DECID ψ → ((¬ φ ψ) ↔ ¬ (φ ¬ ψ))))
 
Theorempm4.56 805 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ φ ¬ ψ) ↔ ¬ (φ ψ))
 
Theoremoranim 806 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.)
((φ ψ) → ¬ (¬ φ ¬ ψ))
 
Theorempm4.78i 807 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.)
(((φψ) (φχ)) → (φ → (ψ χ)))
 
Theorempm4.79dc 808 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 φ → (DECID ψ → (((ψφ) (χφ)) ↔ ((ψ χ) → φ))))
 
Theorempm5.17dc 809 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 ψ → (((φ ψ) ¬ (φ ψ)) ↔ (φ ↔ ¬ ψ)))
 
Theorempm2.85dc 810 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 φ → (((φ ψ) → (φ χ)) → (φ (ψχ))))
 
Theoremorimdidc 811 Disjunction distributes over implication. The forward direction, pm2.76 720, is valid intuitionistically. The reverse direction holds if φ is decidable, as can be seen at pm2.85dc 810. (Contributed by Jim Kingdon, 1-Apr-2018.)
(DECID φ → ((φ (ψχ)) ↔ ((φ ψ) → (φ χ))))
 
Theorempm2.26dc 812 Decidable proposition version of theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.)
(DECID φ → (¬ φ ((φψ) → ψ)))
 
Theorempm4.81dc 813 Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 622 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)
(DECID φ → ((¬ φφ) ↔ φ))
 
Theorempm5.11dc 814 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 φ → (DECID ψ → ((φψ) φψ))))
 
Theorempm5.12dc 815 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 ψ → ((φψ) (φ → ¬ ψ)))
 
Theorempm5.14dc 816 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 ψ → ((φψ) (ψχ)))
 
Theorempm5.13dc 817 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 ψ → ((φψ) (ψφ)))
 
Theorempm5.55dc 818 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 φ → (((φ ψ) ↔ φ) ((φ ψ) ↔ ψ)))
 
Theorempeircedc 819 Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 743, condc 748, or notnotdc 765 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 φ → (((φψ) → φ) → φ))
 
Theoremlooinvdc 820 The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 793, 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 φ → (((φψ) → ψ) → ((ψφ) → φ)))
 
1.2.10  Testable propositions
 
Theoremdftest 821 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 ¬ x = y corresponds to "x = y is testable". (Contributed by David A. Wheeler, 13-Aug-2018.)

(DECID ¬ φ ↔ (¬ φ ¬ ¬ φ))
 
Theoremtestbitestn 822 A proposition is testable iff its negation is testable. See also dcn 745 (which could be read as "Decidability implies testability"). (Contributed by David A. Wheeler, 6-Dec-2018.)
(DECID ¬ φDECID ¬ ¬ φ)
 
Theoremstabtestimpdc 823 "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)
((STAB φ DECID ¬ φ) ↔ DECID φ)
 
1.2.11  Miscellaneous theorems of propositional calculus
 
Theorempm5.21nd 824 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
((φ ψ) → θ)    &   ((φ χ) → θ)    &   (θ → (ψχ))       (φ → (ψχ))
 
Theorempm5.35 825 Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(((φψ) (φχ)) → (φ → (ψχ)))
 
Theorempm5.54dc 826 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 φ → (((φ ψ) ↔ φ) ((φ ψ) ↔ ψ)))
 
Theorembaib 827 Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
(φ ↔ (ψ χ))       (ψ → (φχ))
 
Theorembaibr 828 Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
(φ ↔ (ψ χ))       (ψ → (χφ))
 
Theoremrbaib 829 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(φ ↔ (ψ χ))       (χ → (φψ))
 
Theoremrbaibr 830 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(φ ↔ (ψ χ))       (χ → (ψφ))
 
Theorembaibd 831 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(φ → (ψ ↔ (χ θ)))       ((φ χ) → (ψθ))
 
Theoremrbaibd 832 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(φ → (ψ ↔ (χ θ)))       ((φ θ) → (ψχ))
 
Theorempm5.44 833 Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((φψ) → ((φχ) ↔ (φ → (ψ χ))))
 
Theorempm5.6dc 834 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 835). (Contributed by Jim Kingdon, 2-Apr-2018.)
(DECID ψ → (((φ ¬ ψ) → χ) ↔ (φ → (ψ χ))))
 
Theorempm5.6r 835 Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If ψ is decidable, the converse also holds (see pm5.6dc 834). (Contributed by Jim Kingdon, 4-Aug-2018.)
((φ → (ψ χ)) → ((φ ¬ ψ) → χ))
 
Theoremorcanai 836 Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
(φ → (ψ χ))       ((φ ¬ ψ) → χ)
 
Theoremintnan 837 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
¬ φ        ¬ (ψ φ)
 
Theoremintnanr 838 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
¬ φ        ¬ (φ ψ)
 
Theoremintnand 839 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(φ → ¬ ψ)       (φ → ¬ (χ ψ))
 
Theoremintnanrd 840 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(φ → ¬ ψ)       (φ → ¬ (ψ χ))
 
Theoremdcan 841 A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID φ → (DECID ψDECID (φ ψ)))
 
Theoremdcor 842 A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
(DECID φ → (DECID ψDECID (φ ψ)))
 
Theoremdcbi 843 An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID φ → (DECID ψDECID (φψ)))
 
Theoremannimdc 844 Express conjunction in terms of implication. The forward direction, annimim 781, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)
(DECID φ → (DECID ψ → ((φ ¬ ψ) ↔ ¬ (φψ))))
 
Theorempm4.55dc 845 Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
(DECID φ → (DECID ψ → (¬ (¬ φ ψ) ↔ (φ ¬ ψ))))
 
Theoremmpbiran 846 Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)
ψ    &   (φ ↔ (ψ χ))       (φχ)
 
Theoremmpbiran2 847 Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)
χ    &   (φ ↔ (ψ χ))       (φψ)
 
Theoremmpbir2an 848 Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.) (Revised by NM, 9-Jan-2015.)
ψ    &   χ    &   (φ ↔ (ψ χ))       φ
 
Theoremmpbi2and 849 Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(φψ)    &   (φχ)    &   (φ → ((ψ χ) ↔ θ))       (φθ)
 
Theoremmpbir2and 850 Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(φχ)    &   (φθ)    &   (φ → (ψ ↔ (χ θ)))       (φψ)
 
Theorempm5.62dc 851 Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID ψ → (((φ ψ) ¬ ψ) ↔ (φ ¬ ψ)))
 
Theorempm5.63dc 852 Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID φ → ((φ ψ) ↔ (φ φ ψ))))
 
Theorembianfi 853 A wff conjoined with falsehood is false. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
¬ φ       (φ ↔ (ψ φ))
 
Theorembianfd 854 A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.)
(φ → ¬ ψ)       (φ → (ψ ↔ (ψ χ)))
 
Theorempm4.43 855 Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
(φ ↔ ((φ ψ) (φ ¬ ψ)))
 
Theorempm4.82 856 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
(((φψ) (φ → ¬ ψ)) ↔ ¬ φ)
 
Theorempm4.83dc 857 Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 761, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID φ → (((φψ) φψ)) ↔ ψ))
 
Theorembiantr 858 A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
(((φψ) (χψ)) → (φχ))
 
Theoremorbididc 859 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 φ → ((φ (ψχ)) ↔ ((φ ψ) ↔ (φ χ))))
 
Theorempm5.7dc 860 Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 859. (Contributed by Jim Kingdon, 2-Apr-2018.)
(DECID χ → (((φ χ) ↔ (ψ χ)) ↔ (χ (φψ))))
 
Theorembigolden 861 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
(((φ ψ) ↔ φ) ↔ (ψ ↔ (φ ψ)))
 
Theoremanordc 862 Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 670, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)
(DECID φ → (DECID ψ → ((φ ψ) ↔ ¬ (¬ φ ¬ ψ))))
 
Theorempm3.11dc 863 Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 670, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID φ → (DECID ψ → (¬ (¬ φ ¬ ψ) → (φ ψ))))
 
Theorempm3.12dc 864 Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID φ → (DECID ψ → ((¬ φ ¬ ψ) (φ ψ))))
 
Theorempm3.13dc 865 Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 669, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID φ → (DECID ψ → (¬ (φ ψ) → (¬ φ ¬ ψ))))
 
Theoremdn1dc 866 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 φ (DECID ψ (DECID χ DECID θ))) → (¬ (¬ (¬ (φ ψ) χ) ¬ (φ ¬ (¬ χ ¬ (χ θ)))) ↔ χ))
 
Theorempm5.71dc 867 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 ψ → ((ψ → ¬ χ) → (((φ ψ) χ) ↔ (φ χ))))
 
Theorempm5.75 868 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.)
(((χ → ¬ ψ) (φ ↔ (ψ χ))) → ((φ ¬ ψ) ↔ χ))
 
Theorembimsc1 869 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
(((φψ) (χ ↔ (ψ φ))) → (χφ))
 
Theoremccase 870 Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
((φ ψ) → τ)    &   ((χ ψ) → τ)    &   ((φ θ) → τ)    &   ((χ θ) → τ)       (((φ χ) (ψ θ)) → τ)
 
Theoremccased 871 Deduction for combining cases. (Contributed by NM, 9-May-2004.)
(φ → ((ψ χ) → η))    &   (φ → ((θ χ) → η))    &   (φ → ((ψ τ) → η))    &   (φ → ((θ τ) → η))       (φ → (((ψ θ) (χ τ)) → η))
 
Theoremccase2 872 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
((φ ψ) → τ)    &   (χτ)    &   (θτ)       (((φ χ) (ψ θ)) → τ)
 
Theoremniabn 873 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
φ       ψ → ((χ ψ) ↔ ¬ φ))
 
Theoremdedlem0a 874 Alternate version of dedlema 875. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(φ → (ψ ↔ ((χφ) → (ψ φ))))
 
Theoremdedlema 875 Lemma for iftrue 3330. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(φ → (ψ ↔ ((ψ φ) (χ ¬ φ))))
 
Theoremdedlemb 876 Lemma for iffalse 3333. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
φ → (χ ↔ ((ψ φ) (χ ¬ φ))))
 
Theorempm4.42r 877 One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
(((φ ψ) (φ ¬ ψ)) → φ)
 
Theoremninba 878 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
φ       ψ → (¬ φ ↔ (χ ψ)))
 
Theoremprlem1 879 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.)
(φ → (ηχ))    &   (ψ → ¬ θ)       (φ → (ψ → (((ψ χ) (θ τ)) → η)))
 
Theoremprlem2 880 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.)
(((φ ψ) (χ θ)) ↔ ((φ χ) ((φ ψ) (χ θ))))
 
Theoremoplem1 881 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.)
(φ → (ψ χ))    &   (φ → (θ τ))    &   (ψθ)    &   (χ → (θτ))       (φψ)
 
Theoremrnlem 882 Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((φ ψ) (χ θ)) ↔ (((φ χ) (ψ θ)) ((φ θ) (ψ χ))))
 
1.2.12  Abbreviated conjunction and disjunction of three wff's
 
Syntaxw3o 883 Extend wff definition to include 3-way disjunction ('or').
wff (φ ψ χ)
 
Syntaxw3a 884 Extend wff definition to include 3-way conjunction ('and').
wff (φ ψ χ)
 
Definitiondf-3or 885 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 683. (Contributed by NM, 8-Apr-1994.)
((φ ψ χ) ↔ ((φ ψ) χ))
 
Definitiondf-3an 886 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 381. (Contributed by NM, 8-Apr-1994.)
((φ ψ χ) ↔ ((φ ψ) χ))
 
Theorem3orass 887 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
((φ ψ χ) ↔ (φ (ψ χ)))
 
Theorem3anass 888 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((φ ψ χ) ↔ (φ (ψ χ)))
 
Theorem3anrot 889 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((φ ψ χ) ↔ (ψ χ φ))
 
Theorem3orrot 890 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
((φ ψ χ) ↔ (ψ χ φ))
 
Theorem3ancoma 891 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((φ ψ χ) ↔ (ψ φ χ))
 
Theorem3ancomb 892 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((φ ψ χ) ↔ (φ χ ψ))
 
Theorem3orcomb 893 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
((φ ψ χ) ↔ (φ χ ψ))
 
Theorem3anrev 894 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((φ ψ χ) ↔ (χ ψ φ))
 
Theorem3anan32 895 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
((φ ψ χ) ↔ ((φ χ) ψ))
 
Theorem3anan12 896 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
((φ ψ χ) ↔ (ψ (φ χ)))
 
Theoremanandi3 897 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((φ ψ χ) ↔ ((φ ψ) (φ χ)))
 
Theoremanandi3r 898 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((φ ψ χ) ↔ ((φ ψ) (χ ψ)))
 
Theorem3ioran 899 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
(¬ (φ ψ χ) ↔ (¬ φ ¬ ψ ¬ χ))
 
Theorem3simpa 900 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((φ ψ χ) → (φ ψ))
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