P. 9 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	pm2.64 801	Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑))
Theorem	pm5.53 802	Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((((𝜑 ∨ 𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑 → 𝜃) ∧ (𝜓 → 𝜃)) ∧ (𝜒 → 𝜃)))
Theorem	pm2.38 803	Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
⊢ ((𝜓 → 𝜒) → ((𝜓 ∨ 𝜑) → (𝜒 ∨ 𝜑)))
Theorem	pm2.36 804	Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
⊢ ((𝜓 → 𝜒) → ((𝜑 ∨ 𝜓) → (𝜒 ∨ 𝜑)))
Theorem	pm2.37 805	Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
⊢ ((𝜓 → 𝜒) → ((𝜓 ∨ 𝜑) → (𝜑 ∨ 𝜒)))
Theorem	pm2.73 806	Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → (((𝜑 ∨ 𝜓) ∨ 𝜒) → (𝜓 ∨ 𝜒)))
Theorem	pm2.74 807	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	pm2.76 808	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	pm2.75 809	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	pm2.8 810	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	pm2.81 811	Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜓 → (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜓) → ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜃))))
Theorem	pm2.82 812	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 813	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
⊢ ¬ 𝜑    &   ⊢ ¬ 𝜓    ⇒   ⊢ ¬ (𝜑 ∨ 𝜓)
Theorem	orabs 814	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 815	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 816	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 817	Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))
Theorem	andi 818	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 819	Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
Theorem	orddi 820	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 821	Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
⊢ (((𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ∨ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))))
Theorem	pm4.39 822	Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜃)))
Theorem	animorl 823	Conjunction implies disjunction with one common formula (1/4). (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜒))
Theorem	animorr 824	Conjunction implies disjunction with one common formula (2/4). (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜓))
Theorem	animorlr 825	Conjunction implies disjunction with one common formula (3/4). (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜑))
Theorem	animorrl 826	Conjunction implies disjunction with one common formula (4/4). (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ∧ 𝜓) → (𝜓 ∨ 𝜒))
Theorem	pm4.72 827	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	pm5.16 828	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 829	A disjunction with a true formula is equivalent to that true formula. (Contributed by NM, 23-May-1999.)
⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓)))
1.2.7  Stable propositions
Syntax	wstab 830	Extend wff definition to include stability.
wff STAB 𝜑
Definition	df-stab 831	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 832	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 833	Every negated formula is stable. (Contributed by David A. Wheeler, 13-Aug-2018.)
⊢ STAB ¬ 𝜑
1.2.8  Decidable propositions
Syntax	wdc 834	Extend wff definition to include decidability.
wff DECID 𝜑
Definition	df-dc 835	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 853, exmiddc 836, peircedc 914, or notnotrdc 843, 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 836	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 837	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 838	Equivalence property for decidability. Deduction form. (Contributed by Jim Kingdon, 7-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (DECID 𝜓 ↔ DECID 𝜒))
Theorem	dcbiit 839	Equivalence property for decidability. Closed form. (Contributed by BJ, 27-Jan-2020.)
⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))
Theorem	dcbii 840	Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (DECID 𝜑 ↔ DECID 𝜓)
Theorem	dcim 841	An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
Theorem	dcn 842	The negation of a decidable proposition is decidable. The converse need not hold, but does hold for negated propositions, see dcnn 848. (Contributed by Jim Kingdon, 25-Mar-2018.)
⊢ (DECID 𝜑 → DECID ¬ 𝜑)
Theorem	notnotrdc 843	Double negation elimination for a decidable proposition. The converse, notnot 629, 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 844	Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.)
⊢ (DECID 𝜑 → STAB 𝜑)
Theorem	stdcndc 845	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 846	Obsolete version of stdcndc 845 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 847	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 842. (Contributed by BJ, 18-Nov-2023.)
⊢ (STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑))
Theorem	dcnn 848	Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 842. The relation between dcn 842 and dcnn 848 is analogous to that between notnot 629 and notnotnot 634 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 848 means that a proposition is testable if and only if its negation is testable, and dcn 842 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 849	Obsolete proof of dcnnOLD 849 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 850	Double negation of decidability of a formula. See also comment of nndc 851 to avoid a pitfall that could come from the label "nnexmid". This theorem can also be proved from bj-nnor 14046 as in bj-nndcALT 14070. (Contributed by BJ, 9-Oct-2019.)
⊢ ¬ ¬ (𝜑 ∨ ¬ 𝜑)
Theorem	nndc 851	Double negation of decidability of a formula. Intuitionistic logic refutes the negation of decidability (but does not prove decidability) of any formula. This should not trick the reader into thinking that ¬ ¬ EXMID is provable in intuitionistic logic. Indeed, if we could quantify over formula metavariables, then generalizing nnexmid 850 over 𝜑 would give "⊢ ∀𝜑¬ ¬ DECID 𝜑", but EXMID is "∀𝜑DECID 𝜑", so proving ¬ ¬ EXMID would amount to proving "¬ ¬ ∀𝜑DECID 𝜑", which is not implied by the above theorem. Indeed, the converse of nnal 1647 does not hold. Since our system does not allow quantification over formula metavariables, we can reproduce this argument by representing formulas as subsets of 𝒫 1o, like we do in our definition of EXMID (df-exmid 4190): then, we can prove ∀𝑥 ∈ 𝒫 1o¬ ¬ DECID 𝑥 = 1o but we cannot prove ¬ ¬ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o because the converse of nnral 2465 does not hold. Actually, ¬ ¬ EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying ¬ EXMID and noncontradiction holds (pm3.24 693). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of ¬ ¬ EXMID. (Revised by BJ, 11-Aug-2024.)
⊢ ¬ ¬ DECID 𝜑
1.2.9  Theorems of decidable propositions Many theorems of logic hold in intuitionistic logic just as they do in classical (non-intuitionistic) logic, for all propositions. Other theorems only hold for decidable propositions, such as the law of the excluded middle (df-dc 835), double negation elimination (notnotrdc 843), or contraposition (condc 853). 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 831). Decidable propositions are stable (dcstab 844), but the converse need not hold.
Theorem	const 852	Contraposition when the antecedent is a negated stable proposition. See comment of condc 853. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
⊢ (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
Theorem	condc 853	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 854	Obsolete proof of condc 853 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 855	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 pm2.01 616 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))
Theorem	con1dc 856	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 857	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 858	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 859	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 860	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 861	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 862	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 863	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 864	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 865	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 866	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 867	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 868	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 869	Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. Note that by replacing in proof pm2.52 656 with conax1k 654, 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 870	Alternate proof of pm2.521dc 869. (Contributed by Jim Kingdon, 5-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜓 → 𝜑)))
Theorem	con34bdc 871	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 872	Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 629, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
Theorem	con1biimdc 873	Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑)))
Theorem	con1bidc 874	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑))))
Theorem	con2bidc 875	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))
Theorem	con1biddc 876	A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜒 ↔ 𝜓)))
Theorem	con1biidc 877	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝜓))    ⇒   ⊢ (DECID 𝜑 → (¬ 𝜓 ↔ 𝜑))
Theorem	con1bdc 878	Contraposition. Bidirectional version of con1dc 856. (Contributed by NM, 5-Aug-1993.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑))))
Theorem	con2biidc 879	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
⊢ (DECID 𝜓 → (𝜑 ↔ ¬ 𝜓))    ⇒   ⊢ (DECID 𝜓 → (𝜓 ↔ ¬ 𝜑))
Theorem	con2biddc 880	A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
⊢ (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒)))    ⇒   ⊢ (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓)))
Theorem	condandc 881	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 pm2.65 659, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒)    ⇒   ⊢ (DECID 𝜓 → (𝜑 → 𝜓))
Theorem	bijadc 882	Combine antecedents into a single biconditional. This inference is reminiscent of jadc 863. (Contributed by Jim Kingdon, 4-May-2018.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒))    ⇒   ⊢ (DECID 𝜓 → ((𝜑 ↔ 𝜓) → 𝜒))
Theorem	pm5.18dc 883	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 884	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 637. (Contributed by Jim Kingdon, 30-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))))
Theorem	pm2.13dc 885	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 886	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 887	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 888	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 889	Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 688, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
Theorem	pm4.14dc 890	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 891	Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 722, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))
Theorem	dfordc 892	Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 722, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)
⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))
Theorem	pm2.25dc 893	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 894	Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 748 and one half of dfor2dc 895. (Contributed by Jim Kingdon, 27-Mar-2018.)
⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)))
Theorem	dfor2dc 895	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 896	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 897	Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 721, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)))
Theorem	pm4.62dc 898	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 899	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, pm3.14 753, 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 900	Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 722, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓)))