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Theorem List for Intuitionistic Logic Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm4.64dc 801 Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 641, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜓) ↔ (𝜑𝜓)))

Theorempm4.66dc 802 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 803 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 804 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 805 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))))

Theorempm4.56 806 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theoremoranim 807 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 808 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 809 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 810 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 811 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 812 Disjunction distributes over implication. The forward direction, pm2.76 721, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 811. (Contributed by Jim Kingdon, 1-Apr-2018.)
(DECID 𝜑 → ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒))))

Theorempm2.26dc 813 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 814 Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 623 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜑) ↔ 𝜑))

Theorempm5.11dc 815 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 816 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 817 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 818 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 819 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 820 Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 744, condc 749, or notnotrdc 751 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 821 The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 794, 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 822 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 823 A proposition is testable iff its negation is testable. See also dcn 746 (which could be read as "Decidability implies testability"). (Contributed by David A. Wheeler, 6-Dec-2018.)
(DECID ¬ 𝜑DECID ¬ ¬ 𝜑)

Theoremstabtestimpdc 824 "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 825 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))

Theorempm5.35 826 Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))

Theorempm5.54dc 827 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 828 Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜑𝜒))

Theorembaibr 829 Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))

Theoremrbaib 830 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜑𝜓))

Theoremrbaibr 831 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜓𝜑))

Theorembaibd 832 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜒) → (𝜓𝜃))

Theoremrbaibd 833 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜃) → (𝜓𝜒))

Theorempm5.44 834 Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜑 → (𝜓𝜒))))

Theorempm5.6dc 835 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 836). (Contributed by Jim Kingdon, 2-Apr-2018.)
(DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))

Theorempm5.6r 836 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 835). (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝜑 → (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))

Theoremorcanai 837 Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑 ∧ ¬ 𝜓) → 𝜒)

Theoremintnan 838 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
¬ 𝜑        ¬ (𝜓𝜑)

Theoremintnanr 839 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
¬ 𝜑        ¬ (𝜑𝜓)

Theoremintnand 840 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜓))

Theoremintnanrd 841 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜓𝜒))

Theoremdcan 842 A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))

Theoremdcor 843 A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))

Theoremdcbi 844 An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))

Theoremannimdc 845 Express conjunction in terms of implication. The forward direction, annimim 782, 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 846 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 847 Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)
𝜓    &   (𝜑 ↔ (𝜓𝜒))       (𝜑𝜒)

Theoremmpbiran2 848 Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)
𝜒    &   (𝜑 ↔ (𝜓𝜒))       (𝜑𝜓)

Theoremmpbir2an 849 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 850 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 851 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 852 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 853 Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID 𝜑 → ((𝜑𝜓) ↔ (𝜑 ∨ (¬ 𝜑𝜓))))

Theorembianfi 854 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 855 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)))

Theorempm4.43 856 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)))

Theorempm4.82 857 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)

Theorempm4.83dc 858 Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 762, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
(DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓))

Theorembiantr 859 A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
(((𝜑𝜓) ∧ (𝜒𝜓)) → (𝜑𝜒))

Theoremorbididc 860 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 861 Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 860. (Contributed by Jim Kingdon, 2-Apr-2018.)
(DECID 𝜒 → (((𝜑𝜒) ↔ (𝜓𝜒)) ↔ (𝜒 ∨ (𝜑𝜓))))

Theorembigolden 862 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑𝜓)))

Theoremanordc 863 Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 671, 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 864 Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 671, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))

Theorempm3.12dc 865 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 866 Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 670, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))

Theoremdn1dc 867 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 868 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))))

Theorempm5.75 869 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 870 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
(((𝜑𝜓) ∧ (𝜒 ↔ (𝜓𝜑))) → (𝜒𝜑))

Theoremccase 871 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 872 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 873 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
((𝜑𝜓) → 𝜏)    &   (𝜒𝜏)    &   (𝜃𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)

Theoremniabn 874 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → ((𝜒𝜓) ↔ ¬ 𝜑))

Theoremdedlem0a 875 Alternate version of dedlema 876. (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 876 Lemma for iftrue 3336. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Theoremdedlemb 877 Lemma for iffalse 3339. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Theorempm4.42r 878 One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
(((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)) → 𝜑)

Theoremninba 879 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → (¬ 𝜑 ↔ (𝜒𝜓)))

Theoremprlem1 880 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 881 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 882 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 883 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 884 Extend wff definition to include 3-way disjunction ('or').
wff (𝜑𝜓𝜒)

Syntaxw3a 885 Extend wff definition to include 3-way conjunction ('and').
wff (𝜑𝜓𝜒)

Definitiondf-3or 886 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 684. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))

Definitiondf-3an 887 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.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))

Theorem3orass 888 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))

Theorem3anass 889 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))

Theorem3anrot 890 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))

Theorem3orrot 891 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))

Theorem3ancoma 892 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))

Theorem3ancomb 893 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))

Theorem3orcomb 894 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))

Theorem3anrev 895 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜒𝜓𝜑))

Theorem3anan32 896 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))

Theorem3anan12 897 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
((𝜑𝜓𝜒) ↔ (𝜓 ∧ (𝜑𝜒)))

Theoremanandi3 898 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremanandi3r 899 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theorem3ioran 900 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))

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