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Theorem List for Intuitionistic Logic Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremorordir 701 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))

Theorempm2.3 702 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜑 ∨ (𝜒𝜓)))

Theorempm2.41 703 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜓 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm2.42 704 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm2.4 705 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm4.44 706 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∨ (𝜑𝜓)))

Theoremmtord 707 A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ¬ 𝜓)

Theorempm4.45 708 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∧ (𝜑𝜓)))

Theorempm3.48 709 Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Theoremorim12d 710 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))

Theoremorim1d 711 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → (𝜒𝜃)))

Theoremorim2d 712 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))

Theoremorim2 713 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → (𝜑𝜒)))

Theoremorbi2d 714 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theoremorbi1d 715 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremorbi1 716 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremorbi12d 717 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theorempm5.61 718 Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)
(((𝜑𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Theoremjaoian 719 Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜒)       (((𝜑𝜃) ∧ 𝜓) → 𝜒)

Theoremjao1i 720 Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
(𝜓 → (𝜒𝜑))       ((𝜑𝜓) → (𝜒𝜑))

Theoremjaodan 721 Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)       ((𝜑 ∧ (𝜓𝜃)) → 𝜒)

Theoremmpjaodan 722 Eliminate a disjunction in a deduction. A translation of natural deduction rule E ( elimination). (Contributed by Mario Carneiro, 29-May-2016.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)

Theorempm4.77 723 Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜑) ∧ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))

Theorempm2.63 724 Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((¬ 𝜑𝜓) → 𝜓))

Theorempm2.64 725 Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑))

Theorempm5.53 726 Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((((𝜑𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑𝜃) ∧ (𝜓𝜃)) ∧ (𝜒𝜃)))

Theorempm2.38 727 Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜓𝜑) → (𝜒𝜑)))

Theorempm2.36 728 Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜑𝜓) → (𝜒𝜑)))

Theorempm2.37 729 Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜓𝜑) → (𝜑𝜒)))

Theorempm2.73 730 Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → (((𝜑𝜓) ∨ 𝜒) → (𝜓𝜒)))

Theorempm2.74 731 Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.)
((𝜓𝜑) → (((𝜑𝜓) ∨ 𝜒) → (𝜑𝜒)))

Theorempm2.76 732 Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theorempm2.75 733 Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.)
((𝜑𝜓) → ((𝜑 ∨ (𝜓𝜒)) → (𝜑𝜒)))

Theorempm2.8 734 Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.)
((𝜑𝜓) → ((¬ 𝜓𝜒) → (𝜑𝜒)))

Theorempm2.81 735 Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
((𝜓 → (𝜒𝜃)) → ((𝜑𝜓) → ((𝜑𝜒) → (𝜑𝜃))))

Theorempm2.82 736 Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∨ 𝜒) → (((𝜑 ∨ ¬ 𝜒) ∨ 𝜃) → ((𝜑𝜓) ∨ 𝜃)))

Theorempm3.2ni 737 Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
¬ 𝜑    &    ¬ 𝜓        ¬ (𝜑𝜓)

Theoremorabs 738 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 ↔ ((𝜑𝜓) ∧ 𝜑))

Theoremoranabs 739 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) ↔ (𝜑𝜓))

Theoremordi 740 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 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremordir 741 Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))

Theoremandi 742 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 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))

Theoremandir 743 Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))

Theoremorddi 744 Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))

Theoremanddi 745 Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∨ (𝜑𝜃)) ∨ ((𝜓𝜒) ∨ (𝜓𝜃))))

Theorempm4.39 746 Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜒) ∧ (𝜓𝜃)) → ((𝜑𝜓) ↔ (𝜒𝜃)))

Theorempm4.72 747 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.)
((𝜑𝜓) ↔ (𝜓 ↔ (𝜑𝜓)))

Theorempm5.16 748 Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ ((𝜑𝜓) ∧ (𝜑 ↔ ¬ 𝜓))

Theorembiort 749 A wff is disjoined with truth is true. (Contributed by NM, 23-May-1999.)
(𝜑 → (𝜑 ↔ (𝜑𝜓)))

1.2.7  Stable propositions

Syntaxwstab 750 Extend wff definition to include stability.
wff STAB 𝜑

Definitiondf-stab 751 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 ↔ (¬ ¬ 𝜑𝜑))

Theoremstabnot 752 Every formula of the form ¬ 𝜑 is stable. Uses notnotnot 638. (Contributed by David A. Wheeler, 13-Aug-2018.)
STAB ¬ 𝜑

1.2.8  Decidable propositions

Syntaxwdc 753 Extend wff definition to include decidability.
wff DECID 𝜑

Definitiondf-dc 754 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 760, exmiddc 755, peircedc 831, or notnotrdc 762, any of which would correspond to the assertion that all propositions are decidable.

(Contributed by Jim Kingdon, 11-Mar-2018.)

(DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))

Theoremexmiddc 755 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))

Theorempm2.1dc 756 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 → (¬ 𝜑𝜑))

Theoremdcn 757 A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
(DECID 𝜑DECID ¬ 𝜑)

Theoremdcbii 758 The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
(𝜑𝜓)       (DECID 𝜑DECID 𝜓)

Theoremdcbid 759 The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
(𝜑 → (𝜓𝜒))       (𝜑 → (DECID 𝜓DECID 𝜒))

1.2.9  Theorems of decidable propositions

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 754), double negation elimination (notnotrdc 762), or contraposition (condc 760). 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.

Theoremcondc 760 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) → (𝜓𝜑)))

Theorempm2.18dc 761 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 pm2.01 556 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜑) → 𝜑))

Theoremnotnotrdc 762 Double negation elimination for a decidable proposition. The converse, notnot 569, 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 → (¬ ¬ 𝜑𝜑))

Theoremdcimpstab 763 Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.)
(DECID 𝜑STAB 𝜑)

Theoremcon1dc 764 Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))

Theoremcon4biddc 765 A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.)
(𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 ↔ ¬ 𝜒))))       (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓𝜒))))

Theoremimpidc 766 An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.)
(DECID 𝜒 → (𝜑 → (𝜓𝜒)))       (DECID 𝜒 → (¬ (𝜑 → ¬ 𝜓) → 𝜒))

Theoremsimprimdc 767 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))

Theoremsimplimdc 768 Simplification for a decidable proposition. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 29-Mar-2018.)
(DECID 𝜑 → (¬ (𝜑𝜓) → 𝜑))

Theorempm2.61ddc 769 Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (¬ 𝜓𝜒))       (DECID 𝜓 → (𝜑𝜒))

Theorempm2.6dc 770 Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓)))

Theoremjadc 771 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 → (¬ 𝜑𝜒))    &   (𝜓𝜒)       (DECID 𝜑 → ((𝜑𝜓) → 𝜒))

Theoremjaddc 772 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 → (¬ 𝜓𝜃)))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (DECID 𝜓 → ((𝜓𝜒) → 𝜃)))

Theorempm2.61dc 773 Case elimination for a decidable proposition. Based on theorem *2.61 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.)
(DECID 𝜑 → ((𝜑𝜓) → ((¬ 𝜑𝜓) → 𝜓)))

Theorempm2.5dc 774 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 → (¬ (𝜑𝜓) → (¬ 𝜑𝜓)))

Theorempm2.521dc 775 Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. (Contributed by Jim Kingdon, 5-May-2018.)
(DECID 𝜑 → (¬ (𝜑𝜓) → (𝜓𝜑)))

Theoremcon34bdc 776 Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.)
(DECID 𝜓 → ((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)))

Theoremnotnotbdc 777 Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 569, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
(DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))

Theoremcon1biimdc 778 Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
(DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))

Theoremcon1bidc 779 Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))

Theoremcon2bidc 780 Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))

Theoremcon1biddc 781 A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
(𝜑 → (DECID 𝜓 → (¬ 𝜓𝜒)))       (𝜑 → (DECID 𝜓 → (¬ 𝜒𝜓)))

Theoremcon1biidc 782 A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
(DECID 𝜑 → (¬ 𝜑𝜓))       (DECID 𝜑 → (¬ 𝜓𝜑))

Theoremcon1bdc 783 Contraposition. Bidirectional version of con1dc 764. (Contributed by NM, 5-Aug-1993.)
(DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))

Theoremcon2biidc 784 A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
(DECID 𝜓 → (𝜑 ↔ ¬ 𝜓))       (DECID 𝜓 → (𝜓 ↔ ¬ 𝜑))

Theoremcon2biddc 785 A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
(𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒)))       (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓)))

Theoremcondandc 786 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 pm2.65 595, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
((𝜑 ∧ ¬ 𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒)       (DECID 𝜓 → (𝜑𝜓))

Theorembijadc 787 Combine antecedents into a single biconditional. This inference is reminiscent of jadc 771. (Contributed by Jim Kingdon, 4-May-2018.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (¬ 𝜓𝜒))       (DECID 𝜓 → ((𝜑𝜓) → 𝜒))

Theorempm5.18dc 788 Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider ¬ (𝜑 ↔ ¬ 𝜓) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))

Theoremdfandc 789 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 576. (Contributed by Jim Kingdon, 30-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))))

Theorempm2.13dc 790 A decidable proposition or its triple negation is true. Theorem *2.13 of [WhiteheadRussell] p. 101 with decidability condition added. (Contributed by Jim Kingdon, 13-May-2018.)
(DECID 𝜑 → (𝜑 ∨ ¬ ¬ ¬ 𝜑))

Theorempm4.63dc 791 Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑𝜓))))

Theorempm4.67dc 792 Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑𝜓))))

Theoremannimim 793 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 856. (Contributed by Jim Kingdon, 24-Dec-2017.)
((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))

Theorempm4.65r 794 One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (¬ 𝜑𝜓))

Theoremdcim 795 An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
(DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))

Theoremimanim 796 Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 797. (Contributed by Jim Kingdon, 24-Dec-2017.)
((𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))

Theoremimandc 797 Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 796, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
(DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))

Theorempm4.14dc 798 Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
(DECID 𝜒 → (((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))

Theorempm3.37dc 799 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
(DECID 𝜒 → (((𝜑𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)))

Theorempm4.15 800 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
(((𝜑𝜓) → ¬ 𝜒) ↔ ((𝜓𝜒) → ¬ 𝜑))

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