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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-stab 801 |
Propositions where a double-negative can be removed are called stable.
See Chapter 2 [Moschovakis] p. 2.
Our notation for stability is a connective STAB which we place before the formula in question. For example, STAB 𝑥 = 𝑦 corresponds to "𝑥 = 𝑦 is stable". (Contributed by David A. Wheeler, 13-Aug-2018.) |
⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑)) | ||
Theorem | stbid 802 | The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (STAB 𝜓 ↔ STAB 𝜒)) | ||
Theorem | stabnot 803 | Every negated formula is stable. (Contributed by David A. Wheeler, 13-Aug-2018.) |
⊢ STAB ¬ 𝜑 | ||
Syntax | wdc 804 | Extend wff definition to include decidability. |
wff DECID 𝜑 | ||
Definition | df-dc 805 |
Propositions which are known to be true or false are called decidable.
The (classical) Law of the Excluded Middle corresponds to the principle
that all propositions are decidable, but even given intuitionistic logic,
particular kinds of propositions may be decidable (for example, the
proposition that two natural numbers are equal will be decidable under
most sets of axioms).
Our notation for decidability is a connective DECID which we place before the formula in question. For example, DECID 𝑥 = 𝑦 corresponds to "𝑥 = 𝑦 is decidable". We could transform intuitionistic logic to classical logic by adding unconditional forms of condc 823, exmiddc 806, peircedc 884, or notnotrdc 813, any of which would correspond to the assertion that all propositions are decidable. (Contributed by Jim Kingdon, 11-Mar-2018.) |
⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | ||
Theorem | exmiddc 806 | Law of excluded middle, for a decidable proposition. The law of the excluded middle is also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. The key way in which intuitionistic logic differs from classical logic is that intuitionistic logic says that excluded middle only holds for some propositions, and classical logic says that it holds for all propositions. (Contributed by Jim Kingdon, 12-May-2018.) |
⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | ||
Theorem | pm2.1dc 807 | Commuted law of the excluded middle for a decidable proposition. Based on theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by Jim Kingdon, 25-Mar-2018.) |
⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑)) | ||
Theorem | dcbid 808 | Equivalence property for decidability. Deduction form. (Contributed by Jim Kingdon, 7-Sep-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (DECID 𝜓 ↔ DECID 𝜒)) | ||
Theorem | dcbiit 809 | Equivalence property for decidability. Closed form. (Contributed by BJ, 27-Jan-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓)) | ||
Theorem | dcbii 810 | Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (DECID 𝜑 ↔ DECID 𝜓) | ||
Theorem | dcim 811 | An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓))) | ||
Theorem | dcn 812 | The negation of a decidable proposition is decidable. The converse need not hold, but does hold for negated propositions, see dcnn 818. (Contributed by Jim Kingdon, 25-Mar-2018.) |
⊢ (DECID 𝜑 → DECID ¬ 𝜑) | ||
Theorem | notnotrdc 813 | Double negation elimination for a decidable proposition. The converse, notnot 603, holds for all propositions, not just decidable ones. This is Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability condition added. (Contributed by Jim Kingdon, 11-Mar-2018.) |
⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑)) | ||
Theorem | dcstab 814 | Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.) |
⊢ (DECID 𝜑 → STAB 𝜑) | ||
Theorem | stdcndc 815 | A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.) |
⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑) | ||
Theorem | stdcndcOLD 816 | Obsolete version of stdcndc 815 as of 28-Oct-2023. (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑) | ||
Theorem | stdcn 817 | A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 812. (Contributed by BJ, 18-Nov-2023.) |
⊢ (STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑)) | ||
Theorem | dcnn 818 | Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 812. The relation between dcn 812 and dcnn 818 is analogous to that between notnot 603 and notnotnot 608 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 818 means that a proposition is testable if and only if its negation is testable, and dcn 812 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.) |
⊢ (DECID ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑) | ||
Theorem | dcnnOLD 819 | Obsolete proof of dcnnOLD 819 as of 25-Nov-2023. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (DECID ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑) | ||
Theorem | nnexmid 820 | Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but does not prove excluded middle) for any formula. Can also be proved quickly from bj-nnor 12873 as in bj-nndcALT 12890. (Contributed by BJ, 9-Oct-2019.) |
⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | nndc 821 | Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.) |
⊢ ¬ ¬ DECID 𝜑 | ||
Many theorems of logic hold in intuitionistic logic just as they do in classical (non-inuitionistic) logic, for all propositions. Other theorems only hold for decidable propositions, such as the law of the excluded middle (df-dc 805), double negation elimination (notnotrdc 813), or contraposition (condc 823). Our goal is to prove all well-known or important classical theorems, but with suitable decidability conditions so that the proofs follow from intuitionistic axioms. This section is focused on such proofs, given decidability conditions. Many theorems of this section actually hold for stable propositions (see df-stab 801). Decidable propositions are stable (dcstab 814), but the converse need not hold. | ||
Theorem | const 822 | Contraposition of a stable proposition. See comment of condc 823. (Contributed by BJ, 18-Nov-2023.) |
⊢ (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) | ||
Theorem | condc 823 |
Contraposition of a decidable proposition.
This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) | ||
Theorem | condcOLD 824 | Obsolete proof of condc 823 as of 18-Nov-2023. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) | ||
Theorem | pm2.18dc 825 | Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called Clavius law). Intuitionistically it requires a decidability assumption, but compare with pm2.01 590 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑)) | ||
Theorem | con1dc 826 | Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) | ||
Theorem | con4biddc 827 | A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.) |
⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 ↔ ¬ 𝜒)))) ⇒ ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓 ↔ 𝜒)))) | ||
Theorem | impidc 828 | An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.) |
⊢ (DECID 𝜒 → (𝜑 → (𝜓 → 𝜒))) ⇒ ⊢ (DECID 𝜒 → (¬ (𝜑 → ¬ 𝜓) → 𝜒)) | ||
Theorem | simprimdc 829 | Simplification given a decidable proposition. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 30-Apr-2018.) |
⊢ (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) → 𝜓)) | ||
Theorem | simplimdc 830 | Simplification for a decidable proposition. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 29-Mar-2018.) |
⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → 𝜑)) | ||
Theorem | pm2.61ddc 831 | Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (¬ 𝜓 → 𝜒)) ⇒ ⊢ (DECID 𝜓 → (𝜑 → 𝜒)) | ||
Theorem | pm2.6dc 832 | Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))) | ||
Theorem | jadc 833 | Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.) |
⊢ (DECID 𝜑 → (¬ 𝜑 → 𝜒)) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) → 𝜒)) | ||
Theorem | jaddc 834 | Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.) |
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝜃))) & ⊢ (𝜑 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜒) → 𝜃))) | ||
Theorem | pm2.61dc 835 | Case elimination for a decidable proposition. Theorem *2.61 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.) |
⊢ (DECID 𝜑 → ((𝜑 → 𝜓) → ((¬ 𝜑 → 𝜓) → 𝜓))) | ||
Theorem | pm2.5gdc 836 | Negating an implication for a decidable antecedent. General instance of Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.) |
⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (¬ 𝜑 → 𝜒))) | ||
Theorem | pm2.5dc 837 | Negating an implication for a decidable antecedent. Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.) |
⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (¬ 𝜑 → 𝜓))) | ||
Theorem | pm2.521gdc 838 | A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.) |
⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜒 → 𝜑))) | ||
Theorem | pm2.521dc 839 | Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. Note that by replacing in proof pm2.52 630 with conax1k 628, we obtain a proof of the more general instance where the last occurrence of 𝜑 is replaced with any 𝜒. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜓 → 𝜑))) | ||
Theorem | pm2.521dcALT 840 | Alternate proof of pm2.521dc 839. (Contributed by Jim Kingdon, 5-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜓 → 𝜑))) | ||
Theorem | con34bdc 841 | Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))) | ||
Theorem | notnotbdc 842 | Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 603, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.) |
⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) | ||
Theorem | con1biimdc 843 | Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑))) | ||
Theorem | con1bidc 844 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑)))) | ||
Theorem | con2bidc 845 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))) | ||
Theorem | con1biddc 846 | A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.) |
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜒 ↔ 𝜓))) | ||
Theorem | con1biidc 847 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝜓)) ⇒ ⊢ (DECID 𝜑 → (¬ 𝜓 ↔ 𝜑)) | ||
Theorem | con1bdc 848 | Contraposition. Bidirectional version of con1dc 826. (Contributed by NM, 5-Aug-1993.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑)))) | ||
Theorem | con2biidc 849 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
⊢ (DECID 𝜓 → (𝜑 ↔ ¬ 𝜓)) ⇒ ⊢ (DECID 𝜓 → (𝜓 ↔ ¬ 𝜑)) | ||
Theorem | con2biddc 850 | A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.) |
⊢ (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒))) ⇒ ⊢ (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓))) | ||
Theorem | condandc 851 | 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 pm2.65 633, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒) ⇒ ⊢ (DECID 𝜓 → (𝜑 → 𝜓)) | ||
Theorem | bijadc 852 | Combine antecedents into a single biconditional. This inference is reminiscent of jadc 833. (Contributed by Jim Kingdon, 4-May-2018.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒)) ⇒ ⊢ (DECID 𝜓 → ((𝜑 ↔ 𝜓) → 𝜒)) | ||
Theorem | pm5.18dc 853 | 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 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))) | ||
Theorem | dfandc 854 | 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 611. (Contributed by Jim Kingdon, 30-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)))) | ||
Theorem | pm2.13dc 855 | 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)) | ||
Theorem | pm4.63dc 856 | Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑 ∧ 𝜓)))) | ||
Theorem | pm4.67dc 857 | Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∧ 𝜓)))) | ||
Theorem | imanst 858 | Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.) |
⊢ (STAB 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))) | ||
Theorem | imandc 859 | Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 662, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.) |
⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))) | ||
Theorem | pm4.14dc 860 | Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
⊢ (DECID 𝜒 → (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))) | ||
Theorem | pm2.54dc 861 | Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 696, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))) | ||
Theorem | dfordc 862 | Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 696, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.) |
⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))) | ||
Theorem | pm2.25dc 863 | 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 ∨ ((𝜑 ∨ 𝜓) → 𝜓))) | ||
Theorem | pm2.68dc 864 | Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 722 and one half of dfor2dc 865. (Contributed by Jim Kingdon, 27-Mar-2018.) |
⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓))) | ||
Theorem | dfor2dc 865 | Disjunction 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 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓))) | ||
Theorem | imimorbdc 866 | Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.) |
⊢ (DECID 𝜓 → (((𝜓 → 𝜒) → (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒)))) | ||
Theorem | imordc 867 | Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 695, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))) | ||
Theorem | pm4.62dc 868 | 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))) | ||
Theorem | ianordc 869 | 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, pm3.14 727, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.) |
⊢ (DECID 𝜑 → (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))) | ||
Theorem | pm4.64dc 870 | Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 696, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | pm4.66dc 871 | Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability condition. (Contributed by Jim Kingdon, 2-May-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))) | ||
Theorem | pm4.54dc 872 | 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)))) | ||
Theorem | pm4.79dc 873 | 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 → (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) ↔ ((𝜓 ∧ 𝜒) → 𝜑)))) | ||
Theorem | pm5.17dc 874 | 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))) | ||
Theorem | pm2.85dc 875 | 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 ∨ (𝜓 → 𝜒)))) | ||
Theorem | orimdidc 876 | Disjunction distributes over implication. The forward direction, pm2.76 782, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 875. (Contributed by Jim Kingdon, 1-Apr-2018.) |
⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))) | ||
Theorem | pm2.26dc 877 | Decidable proposition version of theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.) |
⊢ (DECID 𝜑 → (¬ 𝜑 ∨ ((𝜑 → 𝜓) → 𝜓))) | ||
Theorem | pm4.81dc 878 | Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 681 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑)) | ||
Theorem | pm5.11dc 879 | 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 → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓)))) | ||
Theorem | pm5.12dc 880 | 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 → ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓))) | ||
Theorem | pm5.14dc 881 | 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 → ((𝜑 → 𝜓) ∨ (𝜓 → 𝜒))) | ||
Theorem | pm5.13dc 882 | 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 → ((𝜑 → 𝜓) ∨ (𝜓 → 𝜑))) | ||
Theorem | pm5.55dc 883 | 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))) | ||
Theorem | peircedc 884 | Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 806, condc 823, or notnotrdc 813 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 → (((𝜑 → 𝜓) → 𝜑) → 𝜑)) | ||
Theorem | looinvdc 885 | The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 865, 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 → (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → 𝜑))) | ||
Theorem | pm5.21nd 886 | 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 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | ||
Theorem | pm5.35 887 | Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) | ||
Theorem | pm5.54dc 888 | 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))) | ||
Theorem | baib 889 | Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜑 ↔ 𝜒)) | ||
Theorem | baibr 890 | Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜒 ↔ 𝜑)) | ||
Theorem | rbaib 891 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜒 → (𝜑 ↔ 𝜓)) | ||
Theorem | rbaibr 892 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜒 → (𝜓 ↔ 𝜑)) | ||
Theorem | baibd 893 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) | ||
Theorem | rbaibd 894 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒)) | ||
Theorem | pm5.44 895 | Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) → ((𝜑 → 𝜒) ↔ (𝜑 → (𝜓 ∧ 𝜒)))) | ||
Theorem | pm5.6dc 896 | 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 897). (Contributed by Jim Kingdon, 2-Apr-2018.) |
⊢ (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒)))) | ||
Theorem | pm5.6r 897 | 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 896). (Contributed by Jim Kingdon, 4-Aug-2018.) |
⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒)) | ||
Theorem | orcanai 898 | Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) | ||
Theorem | intnan 899 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.) |
⊢ ¬ 𝜑 ⇒ ⊢ ¬ (𝜓 ∧ 𝜑) | ||
Theorem | intnanr 900 | Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.) |
⊢ ¬ 𝜑 ⇒ ⊢ ¬ (𝜑 ∧ 𝜓) |
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