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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pm2.64 701 | Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((φ ∨ ψ) → ((φ ∨ ¬ ψ) → φ)) | ||
Theorem | pm5.53 702 | Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((((φ ∨ ψ) ∨ χ) → θ) ↔ (((φ → θ) ∧ (ψ → θ)) ∧ (χ → θ))) | ||
Theorem | pm2.38 703 | Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((ψ → χ) → ((ψ ∨ φ) → (χ ∨ φ))) | ||
Theorem | pm2.36 704 | Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((ψ → χ) → ((φ ∨ ψ) → (χ ∨ φ))) | ||
Theorem | pm2.37 705 | Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.) |
⊢ ((ψ → χ) → ((ψ ∨ φ) → (φ ∨ χ))) | ||
Theorem | pm2.73 706 | Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ ((φ → ψ) → (((φ ∨ ψ) ∨ χ) → (ψ ∨ χ))) | ||
Theorem | pm2.74 707 | Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) |
⊢ ((ψ → φ) → (((φ ∨ ψ) ∨ χ) → (φ ∨ χ))) | ||
Theorem | pm2.76 708 | Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
⊢ ((φ ∨ (ψ → χ)) → ((φ ∨ ψ) → (φ ∨ χ))) | ||
Theorem | pm2.75 709 | Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.) |
⊢ ((φ ∨ ψ) → ((φ ∨ (ψ → χ)) → (φ ∨ χ))) | ||
Theorem | pm2.8 710 | Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.) |
⊢ ((φ ∨ ψ) → ((¬ ψ ∨ χ) → (φ ∨ χ))) | ||
Theorem | pm2.81 711 | Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ ((ψ → (χ → θ)) → ((φ ∨ ψ) → ((φ ∨ χ) → (φ ∨ θ)))) | ||
Theorem | pm2.82 712 | 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 713 | Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) |
⊢ ¬ φ & ⊢ ¬ ψ ⇒ ⊢ ¬ (φ ∨ ψ) | ||
Theorem | orabs 714 | 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 | orabsOLD 715 | Obsolete proof of orabs 714 as of 28-Feb-2014. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ ↔ ((φ ∨ ψ) ∧ φ)) | ||
Theorem | oranabs 716 | 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 717 | 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 718 | Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((φ ∧ ψ) ∨ χ) ↔ ((φ ∨ χ) ∧ (ψ ∨ χ))) | ||
Theorem | andi 719 | 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 720 | Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((φ ∨ ψ) ∧ χ) ↔ ((φ ∧ χ) ∨ (ψ ∧ χ))) | ||
Theorem | orddi 721 | 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 722 | Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((φ ∨ ψ) ∧ (χ ∨ θ)) ↔ (((φ ∧ χ) ∨ (φ ∧ θ)) ∨ ((ψ ∧ χ) ∨ (ψ ∧ θ)))) | ||
Theorem | pm4.39 723 | Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.) |
⊢ (((φ ↔ χ) ∧ (ψ ↔ θ)) → ((φ ∨ ψ) ↔ (χ ∨ θ))) | ||
Theorem | pm4.72 724 | 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 | pm5.16 725 | 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 726 | A wff is disjoined with truth is true. (Contributed by NM, 23-May-1999.) |
⊢ (φ → (φ ↔ (φ ∨ ψ))) | ||
Syntax | wstab 727 | Extend wff definition to include stability. |
wff STAB φ | ||
Definition | df-stab 728 |
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 | stabnot 729 | Every formula of the form ¬ φ is stable. Uses notnotnot 615. (Contributed by David A. Wheeler, 13-Aug-2018.) |
⊢ STAB ¬ φ | ||
Syntax | wdc 730 | Extend wff definition to include decidability. |
wff DECID φ | ||
Definition | df-dc 731 |
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 737, exmiddc 732, peircedc 808, or notnot2dc 739, 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 732 | 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 733 | 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 | dcn 734 | A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.) |
⊢ (DECID φ → DECID ¬ φ) | ||
Theorem | dcbii 735 | The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (DECID φ ↔ DECID ψ) | ||
Theorem | dcbid 736 | The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (DECID ψ ↔ DECID χ)) | ||
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 731), double negation elimination (notnotdc 754), or contraposition (condc 737). 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. | ||
Theorem | condc 737 |
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.) |
⊢ (DECID φ → ((¬ φ → ¬ ψ) → (ψ → φ))) | ||
Theorem | pm2.18dc 738 | Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called the Law of Clavius). Intuitionistically it requires a decidability assumption, but compare with pm2.01 534 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.) |
⊢ (DECID φ → ((¬ φ → φ) → φ)) | ||
Theorem | notnot2dc 739 | Double negation elimination for a decidable proposition. The converse, notnot1 547, 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 | dcimpstab 740 | Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.) |
⊢ (DECID φ → STAB φ) | ||
Theorem | con1dc 741 | 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 742 | 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 743 | 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 744 | 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 745 | 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 746 | 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 747 | 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 748 | 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 749 | 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 750 | Case elimination for a decidable proposition. Based on theorem *2.61 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.) |
⊢ (DECID φ → ((φ → ψ) → ((¬ φ → ψ) → ψ))) | ||
Theorem | pm2.5dc 751 | Negating an implication for a decidable antecedent. Based on theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.) |
⊢ (DECID φ → (¬ (φ → ψ) → (¬ φ → ψ))) | ||
Theorem | pm2.521dc 752 | Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID φ → (¬ (φ → ψ) → (ψ → φ))) | ||
Theorem | con34bdc 753 | 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 | notnotdc 754 | Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot1 547, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.) |
⊢ (DECID φ → (φ ↔ ¬ ¬ φ)) | ||
Theorem | con1biimdc 755 | Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.) |
⊢ (DECID φ → ((¬ φ ↔ ψ) → (¬ ψ ↔ φ))) | ||
Theorem | con1bidc 756 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((¬ φ ↔ ψ) ↔ (¬ ψ ↔ φ)))) | ||
Theorem | con2bidc 757 | Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((φ ↔ ¬ ψ) ↔ (ψ ↔ ¬ φ)))) | ||
Theorem | con1biddc 758 | A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.) |
⊢ (φ → (DECID ψ → (¬ ψ ↔ χ))) ⇒ ⊢ (φ → (DECID ψ → (¬ χ ↔ ψ))) | ||
Theorem | con1biidc 759 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
⊢ (DECID φ → (¬ φ ↔ ψ)) ⇒ ⊢ (DECID φ → (¬ ψ ↔ φ)) | ||
Theorem | con1bdc 760 | Contraposition. Bidirectional version of con1dc 741. (Contributed by NM, 5-Aug-1993.) |
⊢ (DECID φ → (DECID ψ → ((¬ φ → ψ) ↔ (¬ ψ → φ)))) | ||
Theorem | con2biidc 761 | A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
⊢ (DECID ψ → (φ ↔ ¬ ψ)) ⇒ ⊢ (DECID ψ → (ψ ↔ ¬ φ)) | ||
Theorem | con2biddc 762 | A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.) |
⊢ (φ → (DECID χ → (ψ ↔ ¬ χ))) ⇒ ⊢ (φ → (DECID χ → (χ ↔ ¬ ψ))) | ||
Theorem | condandc 763 | 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 pm2.65 572, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.) |
⊢ ((φ ∧ ¬ ψ) → χ) & ⊢ ((φ ∧ ¬ ψ) → ¬ χ) ⇒ ⊢ (DECID ψ → (φ → ψ)) | ||
Theorem | bijadc 764 | Combine antecedents into a single biconditional. This inference is reminiscent of jadc 748. (Contributed by Jim Kingdon, 4-May-2018.) |
⊢ (φ → (ψ → χ)) & ⊢ (¬ φ → (¬ ψ → χ)) ⇒ ⊢ (DECID ψ → ((φ ↔ ψ) → χ)) | ||
Theorem | pm5.18dc 765 | 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 766 | 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 553. (Contributed by Jim Kingdon, 30-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ)))) | ||
Theorem | pm2.13dc 767 | 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 768 | 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 769 | 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 | annimim 770 | 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 831. (Contributed by Jim Kingdon, 24-Dec-2017.) |
⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ)) | ||
Theorem | pm4.65r 771 | 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) → ¬ (¬ φ → ψ)) | ||
Theorem | dcim 772 | An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.) |
⊢ (DECID φ → (DECID ψ → DECID (φ → ψ))) | ||
Theorem | imanim 773 | Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 774. (Contributed by Jim Kingdon, 24-Dec-2017.) |
⊢ ((φ → ψ) → ¬ (φ ∧ ¬ ψ)) | ||
Theorem | imandc 774 | Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 773, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.) |
⊢ (DECID ψ → ((φ → ψ) ↔ ¬ (φ ∧ ¬ ψ))) | ||
Theorem | pm4.14dc 775 | 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 | pm3.37dc 776 | Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.) |
⊢ (DECID χ → (((φ ∧ ψ) → χ) → ((φ ∧ ¬ χ) → ¬ ψ))) | ||
Theorem | pm4.15 777 | Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) |
⊢ (((φ ∧ ψ) → ¬ χ) ↔ ((ψ ∧ χ) → ¬ φ)) | ||
Theorem | pm2.54dc 778 | Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 628, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.) |
⊢ (DECID φ → ((¬ φ → ψ) → (φ ∨ ψ))) | ||
Theorem | dfordc 779 | Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 628, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.) |
⊢ (DECID φ → ((φ ∨ ψ) ↔ (¬ φ → ψ))) | ||
Theorem | pm2.25dc 780 | 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 781 | Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 654 and one half of dfor2dc 782. (Contributed by Jim Kingdon, 27-Mar-2018.) |
⊢ (DECID φ → (((φ → ψ) → ψ) → (φ ∨ ψ))) | ||
Theorem | dfor2dc 782 | 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 → ((φ ∨ ψ) ↔ ((φ → ψ) → ψ))) | ||
Theorem | imimorbdc 783 | 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 784 | Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 785, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.) |
⊢ (DECID φ → ((φ → ψ) ↔ (¬ φ ∨ ψ))) | ||
Theorem | imorr 785 | 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 784. (Contributed by Jim Kingdon, 21-Jul-2018.) |
⊢ ((¬ φ ∨ ψ) → (φ → ψ)) | ||
Theorem | pm4.62dc 786 | 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 787 | 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, pm3.14 657, 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 | oibabs 788 | Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) |
⊢ (((φ ∨ ψ) → (φ ↔ ψ)) ↔ (φ ↔ ψ)) | ||
Theorem | pm4.64dc 789 | Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 628, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.) |
⊢ (DECID φ → ((¬ φ → ψ) ↔ (φ ∨ ψ))) | ||
Theorem | pm4.66dc 790 | 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 | pm4.52im 791 | 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) → ¬ (¬ φ ∨ ψ)) | ||
Theorem | pm4.53r 792 | One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.) |
⊢ ((¬ φ ∨ ψ) → ¬ (φ ∧ ¬ ψ)) | ||
Theorem | pm4.54dc 793 | 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 | pm4.56 794 | Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ φ ∧ ¬ ψ) ↔ ¬ (φ ∨ ψ)) | ||
Theorem | oranim 795 | 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.) |
⊢ ((φ ∨ ψ) → ¬ (¬ φ ∧ ¬ ψ)) | ||
Theorem | pm4.78i 796 | 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.) |
⊢ (((φ → ψ) ∨ (φ → χ)) → (φ → (ψ ∨ χ))) | ||
Theorem | pm4.79dc 797 | 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 798 | 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 799 | 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 800 | Disjunction distributes over implication. The forward direction, pm2.76 708, is valid intuitionistically. The reverse direction holds if φ is decidable, as can be seen at pm2.85dc 799. (Contributed by Jim Kingdon, 1-Apr-2018.) |
⊢ (DECID φ → ((φ ∨ (ψ → χ)) ↔ ((φ ∨ ψ) → (φ ∨ χ)))) |
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