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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pm2.82 801 | Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑 ∨ 𝜓) ∨ 𝜃))) | ||
Theorem | pm3.2ni 802 | Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) |
⊢ ¬ 𝜑 & ⊢ ¬ 𝜓 ⇒ ⊢ ¬ (𝜑 ∨ 𝜓) | ||
Theorem | orabs 803 | Absorption of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 28-Feb-2014.) |
⊢ (𝜑 ↔ ((𝜑 ∨ 𝜓) ∧ 𝜑)) | ||
Theorem | oranabs 804 | Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.) |
⊢ (((𝜑 ∨ ¬ 𝜓) ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)) | ||
Theorem | ordi 805 | Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) |
⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | ||
Theorem | ordir 806 | Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) | ||
Theorem | andi 807 | Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.) |
⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))) | ||
Theorem | andir 808 | Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))) | ||
Theorem | orddi 809 | Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜃)) ∧ ((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)))) | ||
Theorem | anddi 810 | Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ∨ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)))) | ||
Theorem | pm4.39 811 | Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜃))) | ||
Theorem | pm4.72 812 | Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.) |
⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | pm5.16 813 | Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
⊢ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) | ||
Theorem | biort 814 | A wff is disjoined with truth is true. (Contributed by NM, 23-May-1999.) |
⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓))) | ||
Syntax | wstab 815 | Extend wff definition to include stability. |
wff STAB 𝜑 | ||
Definition | df-stab 816 |
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 817 | 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 818 | Every negated formula is stable. (Contributed by David A. Wheeler, 13-Aug-2018.) |
⊢ STAB ¬ 𝜑 | ||
Syntax | wdc 819 | Extend wff definition to include decidability. |
wff DECID 𝜑 | ||
Definition | df-dc 820 |
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 838, exmiddc 821, peircedc 899, or notnotrdc 828, 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 821 | 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 822 | 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 823 | Equivalence property for decidability. Deduction form. (Contributed by Jim Kingdon, 7-Sep-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (DECID 𝜓 ↔ DECID 𝜒)) | ||
Theorem | dcbiit 824 | Equivalence property for decidability. Closed form. (Contributed by BJ, 27-Jan-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓)) | ||
Theorem | dcbii 825 | Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (DECID 𝜑 ↔ DECID 𝜓) | ||
Theorem | dcim 826 | An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓))) | ||
Theorem | dcn 827 | The negation of a decidable proposition is decidable. The converse need not hold, but does hold for negated propositions, see dcnn 833. (Contributed by Jim Kingdon, 25-Mar-2018.) |
⊢ (DECID 𝜑 → DECID ¬ 𝜑) | ||
Theorem | notnotrdc 828 | Double negation elimination for a decidable proposition. The converse, notnot 618, 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 829 | Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.) |
⊢ (DECID 𝜑 → STAB 𝜑) | ||
Theorem | stdcndc 830 | 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 831 | Obsolete version of stdcndc 830 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 832 | 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 827. (Contributed by BJ, 18-Nov-2023.) |
⊢ (STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑)) | ||
Theorem | dcnn 833 | Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 827. The relation between dcn 827 and dcnn 833 is analogous to that between notnot 618 and notnotnot 623 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 833 means that a proposition is testable if and only if its negation is testable, and dcn 827 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 834 | Obsolete proof of dcnnOLD 834 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 835 | 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 12935 as in bj-nndcALT 12952. (Contributed by BJ, 9-Oct-2019.) |
⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | nndc 836 | 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 820), double negation elimination (notnotrdc 828), or contraposition (condc 838). 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 816). Decidable propositions are stable (dcstab 829), but the converse need not hold. | ||
Theorem | const 837 | Contraposition of a stable proposition. See comment of condc 838. (Contributed by BJ, 18-Nov-2023.) |
⊢ (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))) | ||
Theorem | condc 838 |
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 839 | Obsolete proof of condc 838 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 840 | 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 pm2.01 605 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑)) | ||
Theorem | con1dc 841 | 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 842 | 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 843 | 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 844 | 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 845 | 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 846 | 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 847 | 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 848 | 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 849 | 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 850 | 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 851 | 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 852 | 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 853 | 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 854 | Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. Note that by replacing in proof pm2.52 645 with conax1k 643, 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 855 | Alternate proof of pm2.521dc 854. (Contributed by Jim Kingdon, 5-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜓 → 𝜑))) | ||
Theorem | con34bdc 856 | 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 857 | Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 618, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.) |
⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) | ||
Theorem | con1biimdc 858 | Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑))) | ||
Theorem | con1bidc 859 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑)))) | ||
Theorem | con2bidc 860 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))) | ||
Theorem | con1biddc 861 | A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.) |
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜒 ↔ 𝜓))) | ||
Theorem | con1biidc 862 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝜓)) ⇒ ⊢ (DECID 𝜑 → (¬ 𝜓 ↔ 𝜑)) | ||
Theorem | con1bdc 863 | Contraposition. Bidirectional version of con1dc 841. (Contributed by NM, 5-Aug-1993.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑)))) | ||
Theorem | con2biidc 864 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
⊢ (DECID 𝜓 → (𝜑 ↔ ¬ 𝜓)) ⇒ ⊢ (DECID 𝜓 → (𝜓 ↔ ¬ 𝜑)) | ||
Theorem | con2biddc 865 | A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.) |
⊢ (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒))) ⇒ ⊢ (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓))) | ||
Theorem | condandc 866 | 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 pm2.65 648, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒) ⇒ ⊢ (DECID 𝜓 → (𝜑 → 𝜓)) | ||
Theorem | bijadc 867 | Combine antecedents into a single biconditional. This inference is reminiscent of jadc 848. (Contributed by Jim Kingdon, 4-May-2018.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒)) ⇒ ⊢ (DECID 𝜓 → ((𝜑 ↔ 𝜓) → 𝜒)) | ||
Theorem | pm5.18dc 868 | 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 869 | 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 626. (Contributed by Jim Kingdon, 30-Apr-2018.) |
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)))) | ||
Theorem | pm2.13dc 870 | 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 871 | 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 872 | 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 873 | 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 874 | Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 677, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.) |
⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))) | ||
Theorem | pm4.14dc 875 | 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 876 | Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 711, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))) | ||
Theorem | dfordc 877 | Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 711, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.) |
⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))) | ||
Theorem | pm2.25dc 878 | 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 879 | Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 737 and one half of dfor2dc 880. (Contributed by Jim Kingdon, 27-Mar-2018.) |
⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓))) | ||
Theorem | dfor2dc 880 | 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 881 | 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 882 | Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 710, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))) | ||
Theorem | pm4.62dc 883 | 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 884 | 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, pm3.14 742, 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 885 | Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 711, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | pm4.66dc 886 | 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 887 | 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 888 | 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 889 | 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 890 | 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 891 | Disjunction distributes over implication. The forward direction, pm2.76 797, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 890. (Contributed by Jim Kingdon, 1-Apr-2018.) |
⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))) | ||
Theorem | pm2.26dc 892 | 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 893 | Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 696 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.) |
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑)) | ||
Theorem | pm5.11dc 894 | 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 895 | 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 896 | 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 897 | 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 898 | 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 899 | Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 821, condc 838, or notnotrdc 828 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 900 | The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 880, 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))) |
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