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Theorem List for Intuitionistic Logic Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm2.54dc 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 𝜑 → ((¬ 𝜑𝜓) → (𝜑𝜓)))
 
Theoremdfordc 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 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
 
Theorempm2.25dc 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 𝜑 → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))
 
Theorempm2.68dc 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 𝜑 → (((𝜑𝜓) → 𝜓) → (𝜑𝜓)))
 
Theoremdfor2dc 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 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) → 𝜓)))
 
Theoremimimorbdc 806 Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)
(DECID 𝜓 → (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒))))
 
Theoremimordc 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 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
 
Theoremimorr 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.)
((¬ 𝜑𝜓) → (𝜑𝜓))
 
Theorempm4.62dc 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 𝜑 → ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
 
Theoremianordc 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 𝜑 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
 
Theoremoibabs 811 Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
(((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theorempm4.64dc 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 𝜑 → ((¬ 𝜑𝜓) ↔ (𝜑𝜓)))
 
Theorempm4.66dc 813 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 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.)
((𝜑 ∧ ¬ 𝜓) → ¬ (¬ 𝜑𝜓))
 
Theorempm4.53r 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.)
((¬ 𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
 
Theorempm4.54dc 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 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
 
Theorempm4.56 817 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremoranim 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.)
((𝜑𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
 
Theorempm4.78i 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.)
(((𝜑𝜓) ∨ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
 
Theorempm4.79dc 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 𝜑 → (DECID 𝜓 → (((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))))
 
Theorempm5.17dc 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 𝜓 → (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))
 
Theorempm2.85dc 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 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
 
Theoremorimdidc 823 Disjunction distributes over implication. The forward direction, pm2.76 732, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 822. (Contributed by Jim Kingdon, 1-Apr-2018.)
(DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))
 
Theorempm2.26dc 824 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 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 𝜑 → ((¬ 𝜑𝜑) ↔ 𝜑))
 
Theorempm5.11dc 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 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ∨ (¬ 𝜑𝜓))))
 
Theorempm5.12dc 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 𝜓 → ((𝜑𝜓) ∨ (𝜑 → ¬ 𝜓)))
 
Theorempm5.14dc 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 𝜓 → ((𝜑𝜓) ∨ (𝜓𝜒)))
 
Theorempm5.13dc 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 𝜓 → ((𝜑𝜓) ∨ (𝜓𝜑)))
 
Theorempm5.55dc 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 𝜑 → (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓)))
 
Theorempeircedc 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 𝜑 → (((𝜑𝜓) → 𝜑) → 𝜑))
 
Theoremlooinvdc 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 𝜑 → (((𝜑𝜓) → 𝜓) → ((𝜓𝜑) → 𝜑)))
 
1.2.10  Testable propositions
 
Theoremdftest 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 ¬ 𝑥 = 𝑦 corresponds to "x = y is testable". (Contributed by David A. Wheeler, 13-Aug-2018.)

(DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
 
Theoremtestbitestn 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 ¬ 𝜑DECID ¬ ¬ 𝜑)
 
Theoremstabtestimpdc 835 "Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)
((STAB 𝜑DECID ¬ 𝜑) ↔ DECID 𝜑)
 
1.2.11  Miscellaneous theorems of propositional calculus
 
Theorempm5.21nd 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.)
((𝜑𝜓) → 𝜃)    &   ((𝜑𝜒) → 𝜃)    &   (𝜃 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))
 
Theorempm5.35 837 Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
 
Theorempm5.54dc 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 𝜑 → (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓)))
 
Theorembaib 839 Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜑𝜒))
 
Theorembaibr 840 Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theoremrbaib 841 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜑𝜓))
 
Theoremrbaibr 842 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜓𝜑))
 
Theorembaibd 843 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜒) → (𝜓𝜃))
 
Theoremrbaibd 844 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜃) → (𝜓𝜒))
 
Theorempm5.44 845 Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜑 → (𝜓𝜒))))
 
Theorempm5.6dc 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 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
 
Theorempm5.6r 847 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 846). (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝜑 → (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
 
Theoremorcanai 848 Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑 ∧ ¬ 𝜓) → 𝜒)
 
Theoremintnan 849 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
¬ 𝜑        ¬ (𝜓𝜑)
 
Theoremintnanr 850 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
¬ 𝜑        ¬ (𝜑𝜓)
 
Theoremintnand 851 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜓))
 
Theoremintnanrd 852 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜓𝜒))
 
Theoremdcan 853 A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
 
Theoremdcor 854 A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
 
Theoremdcbi 855 An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
 
Theoremannimdc 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 𝜑 → (DECID 𝜓 → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
 
Theorempm4.55dc 857 Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))
 
Theoremmpbiran 858 Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)
𝜓    &   (𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)
 
Theoremmpbiran2 859 Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)
𝜒    &   (𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)
 
Theoremmpbir2an 860 Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.) (Revised by NM, 9-Jan-2015.)
𝜓    &   𝜒    &   (𝜑 ↔ (𝜓𝜒))       𝜑
 
Theoremmpbi2and 861 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 → ((𝜓𝜒) ↔ 𝜃))       (𝜑𝜃)
 
Theoremmpbir2and 862 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 ↔ (𝜒𝜃)))       (𝜑𝜓)
 
Theorempm5.62dc 863 Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID 𝜓 → (((𝜑𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
 
Theorempm5.63dc 864 Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID 𝜑 → ((𝜑𝜓) ↔ (𝜑 ∨ (¬ 𝜑𝜓))))
 
Theorembianfi 865 A wff conjoined with falsehood is false. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
¬ 𝜑       (𝜑 ↔ (𝜓𝜑))
 
Theorembianfd 866 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 ↔ (𝜓𝜒)))
 
Theorempm4.43 867 Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
(𝜑 ↔ ((𝜑𝜓) ∧ (𝜑 ∨ ¬ 𝜓)))
 
Theorempm4.82 868 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)
 
Theorempm4.83dc 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 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓))
 
Theorembiantr 870 A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
(((𝜑𝜓) ∧ (𝜒𝜓)) → (𝜑𝜒))
 
Theoremorbididc 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 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒))))
 
Theorempm5.7dc 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 𝜒 → (((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓))))
 
Theorembigolden 873 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑𝜓)))
 
Theoremanordc 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 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
 
Theorempm3.11dc 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 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
 
Theorempm3.12dc 876 Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))))
 
Theorempm3.13dc 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 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))
 
Theoremdn1dc 878 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 𝜒DECID 𝜃))) → (¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒))
 
Theorempm5.71dc 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 𝜓 → ((𝜓 → ¬ 𝜒) → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒))))
 
Theorempm5.75 880 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.)
(((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
 
Theorembimsc1 881 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
(((𝜑𝜓) ∧ (𝜒 ↔ (𝜓𝜑))) → (𝜒𝜑))
 
Theoremccase 882 Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
((𝜑𝜓) → 𝜏)    &   ((𝜒𝜓) → 𝜏)    &   ((𝜑𝜃) → 𝜏)    &   ((𝜒𝜃) → 𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 
Theoremccased 883 Deduction for combining cases. (Contributed by NM, 9-May-2004.)
(𝜑 → ((𝜓𝜒) → 𝜂))    &   (𝜑 → ((𝜃𝜒) → 𝜂))    &   (𝜑 → ((𝜓𝜏) → 𝜂))    &   (𝜑 → ((𝜃𝜏) → 𝜂))       (𝜑 → (((𝜓𝜃) ∧ (𝜒𝜏)) → 𝜂))
 
Theoremccase2 884 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
((𝜑𝜓) → 𝜏)    &   (𝜒𝜏)    &   (𝜃𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 
Theoremniabn 885 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → ((𝜒𝜓) ↔ ¬ 𝜑))
 
Theoremdedlem0a 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.)
(𝜑 → (𝜓 ↔ ((𝜒𝜑) → (𝜓𝜑))))
 
Theoremdedlema 887 Lemma for iftrue 3363. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 
Theoremdedlemb 888 Lemma for iffalse 3366. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 
Theorempm4.42r 889 One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
(((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)) → 𝜑)
 
Theoremninba 890 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → (¬ 𝜑 ↔ (𝜒𝜓)))
 
Theoremprlem1 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.)
(𝜑 → (𝜂𝜒))    &   (𝜓 → ¬ 𝜃)       (𝜑 → (𝜓 → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂)))
 
Theoremprlem2 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.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
 
Theoremoplem1 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.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜓𝜃)    &   (𝜒 → (𝜃𝜏))       (𝜑𝜓)
 
Theoremrnlem 894 Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜓𝜃)) ∧ ((𝜑𝜃) ∧ (𝜓𝜒))))
 
1.2.12  Abbreviated conjunction and disjunction of three wff's
 
Syntaxw3o 895 Extend wff definition to include 3-way disjunction ('or').
wff (𝜑𝜓𝜒)
 
Syntaxw3a 896 Extend wff definition to include 3-way conjunction ('and').
wff (𝜑𝜓𝜒)
 
Definitiondf-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.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
 
Definitiondf-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.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
 
Theorem3orass 899 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))
 
Theorem3anass 900 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
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