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Theorem List for Intuitionistic Logic Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcon1biimdc 801 Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
 
Theoremcon1bidc 802 Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))
 
Theoremcon2bidc 803 Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))
 
Theoremcon1biddc 804 A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
(𝜑 → (DECID 𝜓 → (¬ 𝜓𝜒)))       (𝜑 → (DECID 𝜓 → (¬ 𝜒𝜓)))
 
Theoremcon1biidc 805 A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
(DECID 𝜑 → (¬ 𝜑𝜓))       (DECID 𝜑 → (¬ 𝜓𝜑))
 
Theoremcon1bdc 806 Contraposition. Bidirectional version of con1dc 787. (Contributed by NM, 5-Aug-1993.)
(DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))
 
Theoremcon2biidc 807 A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
(DECID 𝜓 → (𝜑 ↔ ¬ 𝜓))       (DECID 𝜓 → (𝜓 ↔ ¬ 𝜑))
 
Theoremcon2biddc 808 A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
(𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒)))       (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓)))
 
Theoremcondandc 809 Proof by contradiction. This only holds for decidable propositions, as it is part of the family of theorems which assume ¬ 𝜓, derive a contradiction, and therefore conclude 𝜓. By contrast, assuming 𝜓, deriving a contradiction, and therefore concluding ¬ 𝜓, as in pm2.65 618, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
((𝜑 ∧ ¬ 𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒)       (DECID 𝜓 → (𝜑𝜓))
 
Theorembijadc 810 Combine antecedents into a single biconditional. This inference is reminiscent of jadc 794. (Contributed by Jim Kingdon, 4-May-2018.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (¬ 𝜓𝜒))       (DECID 𝜓 → ((𝜑𝜓) → 𝜒))
 
Theorempm5.18dc 811 Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider ¬ (𝜑 ↔ ¬ 𝜓) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
 
Theoremdfandc 812 Definition of 'and' in terms of negation and implication, for decidable propositions. The forward direction holds for all propositions, and can (basically) be found at pm3.2im 599. (Contributed by Jim Kingdon, 30-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))))
 
Theorempm2.13dc 813 A decidable proposition or its triple negation is true. Theorem *2.13 of [WhiteheadRussell] p. 101 with decidability condition added. (Contributed by Jim Kingdon, 13-May-2018.)
(DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑))
 
Theorempm4.63dc 814 Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑𝜓))))
 
Theorempm4.67dc 815 Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑𝜓))))
 
Theoremannimim 816 Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 879. (Contributed by Jim Kingdon, 24-Dec-2017.)
((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
 
Theorempm4.65r 817 One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (¬ 𝜑𝜓))
 
Theoremdcim 818 An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
 
Theoremimanim 819 Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 820. (Contributed by Jim Kingdon, 24-Dec-2017.)
((𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
 
Theoremimandc 820 Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 819, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
(DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
 
Theorempm4.14dc 821 Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
(DECID 𝜒 → (((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))
 
Theorempm3.37 822 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
 
Theorempm4.15 823 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
(((𝜑𝜓) → ¬ 𝜒) ↔ ((𝜓𝜒) → ¬ 𝜑))
 
Theorempm2.54dc 824 Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 674, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜓) → (𝜑𝜓)))
 
Theoremdfordc 825 Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 674, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)
(DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
 
Theorempm2.25dc 826 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 827 Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 700 and one half of dfor2dc 828. (Contributed by Jim Kingdon, 27-Mar-2018.)
(DECID 𝜑 → (((𝜑𝜓) → 𝜓) → (𝜑𝜓)))
 
Theoremdfor2dc 828 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 829 Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)
(DECID 𝜓 → (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒))))
 
Theoremimordc 830 Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 831, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
(DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
 
Theoremimorr 831 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 830. (Contributed by Jim Kingdon, 21-Jul-2018.)
((¬ 𝜑𝜓) → (𝜑𝜓))
 
Theorempm4.62dc 832 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 833 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, pm3.14 703, 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 834 Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
(((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theorempm4.64dc 835 Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 674, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜓) ↔ (𝜑𝜓)))
 
Theorempm4.66dc 836 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 837 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 838 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 839 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))))
 
Theorempm4.56 840 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremoranim 841 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 842 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 843 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 844 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 845 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 846 Disjunction distributes over implication. The forward direction, pm2.76 755, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 845. (Contributed by Jim Kingdon, 1-Apr-2018.)
(DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))
 
Theorempm2.26dc 847 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 848 Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 656 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜑) ↔ 𝜑))
 
Theorempm5.11dc 849 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 850 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 851 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 852 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 853 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 854 Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 778, condc 783, or notnotrdc 785 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 855 The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 828, 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 856 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 857 A proposition is testable iff its negation is testable. See also dcn 780 (which could be read as "Decidability implies testability"). (Contributed by David A. Wheeler, 6-Dec-2018.)
(DECID ¬ 𝜑DECID ¬ ¬ 𝜑)
 
Theoremstabtestimpdc 858 "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 859 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))
 
Theorempm5.35 860 Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
 
Theorempm5.54dc 861 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 862 Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜑𝜒))
 
Theorembaibr 863 Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theoremrbaib 864 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜑𝜓))
 
Theoremrbaibr 865 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜓𝜑))
 
Theorembaibd 866 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜒) → (𝜓𝜃))
 
Theoremrbaibd 867 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜃) → (𝜓𝜒))
 
Theorempm5.44 868 Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜑 → (𝜓𝜒))))
 
Theorempm5.6dc 869 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 870). (Contributed by Jim Kingdon, 2-Apr-2018.)
(DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
 
Theorempm5.6r 870 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 869). (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝜑 → (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
 
Theoremorcanai 871 Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑 ∧ ¬ 𝜓) → 𝜒)
 
Theoremintnan 872 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
¬ 𝜑        ¬ (𝜓𝜑)
 
Theoremintnanr 873 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
¬ 𝜑        ¬ (𝜑𝜓)
 
Theoremintnand 874 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜓))
 
Theoremintnanrd 875 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜓𝜒))
 
Theoremdcan 876 A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
 
Theoremdcor 877 A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
 
Theoremdcbi 878 An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
 
Theoremannimdc 879 Express conjunction in terms of implication. The forward direction, annimim 816, 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 880 Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))
 
Theoremorandc 881 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 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))
 
Theoremmpbiran 882 Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)
𝜓    &   (𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)
 
Theoremmpbiran2 883 Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)
𝜒    &   (𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)
 
Theoremmpbir2an 884 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 885 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 886 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 887 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 888 Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID 𝜑 → ((𝜑𝜓) ↔ (𝜑 ∨ (¬ 𝜑𝜓))))
 
Theorembianfi 889 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 890 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)))
 
Theorempm4.43 891 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)))
 
Theorempm4.82 892 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)
 
Theorempm4.83dc 893 Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 796, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓))
 
Theorembiantr 894 A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
(((𝜑𝜓) ∧ (𝜒𝜓)) → (𝜑𝜒))
 
Theoremorbididc 895 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 896 Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 895. (Contributed by Jim Kingdon, 2-Apr-2018.)
(DECID 𝜒 → (((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓))))
 
Theorembigolden 897 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑𝜓)))
 
Theoremanordc 898 Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 704, 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 899 Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 704, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
 
Theorempm3.12dc 900 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))))
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