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	orim2 801	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) )
Theorem	orbi2d 802	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.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th \/ ps ) <-> ( th \/ ch ) ) )
Theorem	orbi1d 803	Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ th ) ) )
Theorem	orbi1 804	Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
Theorem	orbi12d 805	Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) )
Theorem	pm5.61 806	Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)
|- ( ( ( ph \/ ps ) /\ -. ps ) <-> ( ph /\ -. ps ) )
Theorem	jaoian 807	Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( th /\ ps ) -> ch )   =>    |- ( ( ( ph \/ th ) /\ ps ) -> ch )
Theorem	jao1i 808	Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
|- ( ps -> ( ch -> ph ) )   =>    |- ( ( ph \/ ps ) -> ( ch -> ph ) )
Theorem	jaodan 809	Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ th ) -> ch )   =>    |- ( ( ph /\ ( ps \/ th ) ) -> ch )
Theorem	mpjaodan 810	Eliminate a disjunction in a deduction. A translation of natural deduction rule \/ E ( \/ elimination). (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ th ) -> ch )   &    |- ( ph -> ( ps \/ th ) )   =>    |- ( ph -> ch )
Theorem	pm4.77 811	Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) <-> ( ( ps \/ ch ) -> ph ) )
Theorem	pm2.63 812	Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( -. ph \/ ps ) -> ps ) )
Theorem	pm2.64 813	Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( ph \/ -. ps ) -> ph ) )
Theorem	pm5.53 814	Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ( ph \/ ps ) \/ ch ) -> th ) <-> ( ( ( ph -> th ) /\ ( ps -> th ) ) /\ ( ch -> th ) ) )
Theorem	pm2.38 815	Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ch \/ ph ) ) )
Theorem	pm2.36 816	Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ch \/ ph ) ) )
Theorem	pm2.37 817	Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ph \/ ch ) ) )
Theorem	pm2.73 818	Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph -> ps ) -> ( ( ( ph \/ ps ) \/ ch ) -> ( ps \/ ch ) ) )
Theorem	pm2.74 819	Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.)
|- ( ( ps -> ph ) -> ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ch ) ) )
Theorem	pm2.76 820	Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( ph \/ ( ps -> ch ) ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) )
Theorem	pm2.75 821	Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.)
|- ( ( ph \/ ps ) -> ( ( ph \/ ( ps -> ch ) ) -> ( ph \/ ch ) ) )
Theorem	pm2.8 822	Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.)
|- ( ( ph \/ ps ) -> ( ( -. ps \/ ch ) -> ( ph \/ ch ) ) )
Theorem	pm2.81 823	Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ( ch -> th ) ) -> ( ( ph \/ ps ) -> ( ( ph \/ ch ) -> ( ph \/ th ) ) ) )
Theorem	pm2.82 824	Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph \/ ps ) \/ ch ) -> ( ( ( ph \/ -. ch ) \/ th ) -> ( ( ph \/ ps ) \/ th ) ) )
Theorem	pm3.2ni 825	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
|- -. ph   &    |- -. ps   =>    |- -. ( ph \/ ps )
Theorem	orabs 826	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.)
|- ( ph <-> ( ( ph \/ ps ) /\ ph ) )
Theorem	oranabs 827	Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.)
|- ( ( ( ph \/ -. ps ) /\ ps ) <-> ( ph /\ ps ) )
Theorem	ordi 828	Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) )
Theorem	ordir 829	Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph /\ ps ) \/ ch ) <-> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )
Theorem	andi 830	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.)
|- ( ( ph /\ ( ps \/ ch ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
Theorem	andir 831	Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )
Theorem	orddi 832	Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <-> ( ( ( ph \/ ch ) /\ ( ph \/ th ) ) /\ ( ( ps \/ ch ) /\ ( ps \/ th ) ) ) )
Theorem	anddi 833	Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) /\ ( ch \/ th ) ) <-> ( ( ( ph /\ ch ) \/ ( ph /\ th ) ) \/ ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) )
Theorem	pm4.39 834	Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) -> ( ( ph \/ ps ) <-> ( ch \/ th ) ) )
Theorem	animorl 835	Conjunction implies disjunction with one common formula (1/4). (Contributed by BJ, 4-Oct-2019.)
|- ( ( ph /\ ps ) -> ( ph \/ ch ) )
Theorem	animorr 836	Conjunction implies disjunction with one common formula (2/4). (Contributed by BJ, 4-Oct-2019.)
|- ( ( ph /\ ps ) -> ( ch \/ ps ) )
Theorem	animorlr 837	Conjunction implies disjunction with one common formula (3/4). (Contributed by BJ, 4-Oct-2019.)
|- ( ( ph /\ ps ) -> ( ch \/ ph ) )
Theorem	animorrl 838	Conjunction implies disjunction with one common formula (4/4). (Contributed by BJ, 4-Oct-2019.)
|- ( ( ph /\ ps ) -> ( ps \/ ch ) )
Theorem	pm4.72 839	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.)
|- ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) )
Theorem	pm5.16 840	Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- -. ( ( ph <-> ps ) /\ ( ph <-> -. ps ) )
Theorem	biort 841	A disjunction with a true formula is equivalent to that true formula. (Contributed by NM, 23-May-1999.)
|- ( ph -> ( ph <-> ( ph \/ ps ) ) )
1.2.7  Stable propositions
Syntax	wstab 842	Extend wff definition to include stability.
wff STAB ph
Definition	df-stab 843	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 ph <-> ( -. -. ph -> ph ) )
Theorem	stbid 844	The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> (STAB ps <-> STAB ch ) )
Theorem	stabnot 845	Every negated formula is stable. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- STAB -. ph
1.2.8  Decidable propositions
Syntax	wdc 846	Extend wff definition to include decidability.
wff DECID ph
Definition	df-dc 847	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 865, exmiddc 848, peircedc 926, or notnotrdc 855, any of which would correspond to the assertion that all propositions are decidable. (Contributed by Jim Kingdon, 11-Mar-2018.)
|- (DECID ph <-> ( ph \/ -. ph ) )
Theorem	exmiddc 848	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 ph -> ( ph \/ -. ph ) )
Theorem	pm2.1dc 849	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 ph -> ( -. ph \/ ph ) )
Theorem	dcbid 850	Equivalence property for decidability. Deduction form. (Contributed by Jim Kingdon, 7-Sep-2019.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> (DECID ps <-> DECID ch ) )
Theorem	dcbiit 851	Equivalence property for decidability. Closed form. (Contributed by BJ, 27-Jan-2020.)
|- ( ( ph <-> ps ) -> (DECID ph <-> DECID ps ) )
Theorem	dcbii 852	Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- ( ph <-> ps )   =>    |- (DECID ph <-> DECID ps )
Theorem	dcim 853	An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
Theorem	dcn 854	The negation of a decidable proposition is decidable. The converse need not hold, but does hold for negated propositions, see dcnn 860. (Contributed by Jim Kingdon, 25-Mar-2018.)
|- (DECID ph -> DECID -. ph )
Theorem	notnotrdc 855	Double negation elimination for a decidable proposition. The converse, notnot 638, 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 ph -> ( -. -. ph -> ph ) )
Theorem	dcstab 856	Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- (DECID ph -> STAB ph )
Theorem	stdcndc 857	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 ph /\ DECID -. ph ) <-> DECID ph )
Theorem	stdcndcOLD 858	Obsolete version of stdcndc 857 as of 28-Oct-2023. (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( (STAB ph /\ DECID -. ph ) <-> DECID ph )
Theorem	stdcn 859	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 854. (Contributed by BJ, 18-Nov-2023.)
|- (STAB ph <-> (DECID -. ph -> DECID ph ) )
Theorem	dcnn 860	Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 854. The relation between dcn 854 and dcnn 860 is analogous to that between notnot 638 and notnotnot 643 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 860 means that a proposition is testable if and only if its negation is testable, and dcn 854 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
|- (DECID -. ph <-> DECID -. -. ph )
Theorem	dcnnOLD 861	Obsolete proof of dcnnOLD 861 as of 25-Nov-2023. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- (DECID -. ph <-> DECID -. -. ph )
Theorem	nnexmid 862	Double negation of decidability of a formula. See also comment of nndc 863 to avoid a pitfall that could come from the label "nnexmid". This theorem can also be proved from bj-nnor 16676 as in bj-nndcALT 16700. (Contributed by BJ, 9-Oct-2019.)
|- -. -. ( ph \/ -. ph )
Theorem	nndc 863	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 862 over ph would give " |- A. ph -. -. DECID ph", but EXMID is " A. phDECID ph", so proving -. -. EXMID would amount to proving " -. -. A. phDECID ph", which is not implied by the above theorem. Indeed, the converse of nnal 1702 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 4327): then, we can prove A. x e. ~P 1o -. -. DECID x = 1o but we cannot prove -. -. A. x e. ~P 1oDECID x = 1o because the converse of nnral 2540 does not hold. Actually, -. -. EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying -. EXMID and noncontradiction holds (pm3.24 705). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of -. -. EXMID. (Revised by BJ, 11-Aug-2024.)
|- -. -. DECID ph
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 847), double negation elimination (notnotrdc 855), or contraposition (condc 865). 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 843). Decidable propositions are stable (dcstab 856), but the converse need not hold.
Theorem	const 864	Contraposition when the antecedent is a negated stable proposition. See comment of condc 865. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
|- (STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
Theorem	condc 865	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 ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
Theorem	condcOLD 866	Obsolete proof of condc 865 as of 18-Nov-2023. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- (DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
Theorem	pm2.18dc 867	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 pm2.01 625 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
|- (DECID ph -> ( ( -. ph -> ph ) -> ph ) )
Theorem	con1dc 868	Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.)
|- (DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
Theorem	con4biddc 869	A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.)
|- ( ph -> (DECID ps -> (DECID ch -> ( -. ps <-> -. ch ) ) ) )   =>    |- ( ph -> (DECID ps -> (DECID ch -> ( ps <-> ch ) ) ) )
Theorem	impidc 870	An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.)
|- (DECID ch -> ( ph -> ( ps -> ch ) ) )   =>    |- (DECID ch -> ( -. ( ph -> -. ps ) -> ch ) )
Theorem	simprimdc 871	Simplification given a decidable proposition. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 30-Apr-2018.)
|- (DECID ps -> ( -. ( ph -> -. ps ) -> ps ) )
Theorem	simplimdc 872	Simplification for a decidable proposition. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 29-Mar-2018.)
|- (DECID ph -> ( -. ( ph -> ps ) -> ph ) )
Theorem	pm2.61ddc 873	Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( -. ps -> ch ) )   =>    |- (DECID ps -> ( ph -> ch ) )
Theorem	pm2.6dc 874	Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.)
|- (DECID ph -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) )
Theorem	jadc 875	Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
|- (DECID ph -> ( -. ph -> ch ) )   &    |- ( ps -> ch )   =>    |- (DECID ph -> ( ( ph -> ps ) -> ch ) )
Theorem	jaddc 876	Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)
|- ( ph -> (DECID ps -> ( -. ps -> th ) ) )   &    |- ( ph -> ( ch -> th ) )   =>    |- ( ph -> (DECID ps -> ( ( ps -> ch ) -> th ) ) )
Theorem	pm2.61dc 877	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 ph -> ( ( ph -> ps ) -> ( ( -. ph -> ps ) -> ps ) ) )
Theorem	pm2.5gdc 878	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 ph -> ( -. ( ph -> ps ) -> ( -. ph -> ch ) ) )
Theorem	pm2.5dc 879	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 ph -> ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) )
Theorem	pm2.521gdc 880	A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.)
|- (DECID ph -> ( -. ( ph -> ps ) -> ( ch -> ph ) ) )
Theorem	pm2.521dc 881	Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. Note that by replacing in proof pm2.52 666 with conax1k 664, we obtain a proof of the more general instance where the last occurrence of ph is replaced with any ch. (Contributed by Jim Kingdon, 5-May-2018.)
|- (DECID ph -> ( -. ( ph -> ps ) -> ( ps -> ph ) ) )
Theorem	pm2.521dcALT 882	Alternate proof of pm2.521dc 881. (Contributed by Jim Kingdon, 5-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- (DECID ph -> ( -. ( ph -> ps ) -> ( ps -> ph ) ) )
Theorem	con34bdc 883	Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.)
|- (DECID ps -> ( ( ph -> ps ) <-> ( -. ps -> -. ph ) ) )
Theorem	notnotbdc 884	Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 638, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
|- (DECID ph -> ( ph <-> -. -. ph ) )
Theorem	con1biimdc 885	Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
|- (DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps <-> ph ) ) )
Theorem	con1bidc 886	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) ) )
Theorem	con2bidc 887	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) )
Theorem	con1biddc 888	A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
|- ( ph -> (DECID ps -> ( -. ps <-> ch ) ) )   =>    |- ( ph -> (DECID ps -> ( -. ch <-> ps ) ) )
Theorem	con1biidc 889	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID ph -> ( -. ph <-> ps ) )   =>    |- (DECID ph -> ( -. ps <-> ph ) )
Theorem	con1bdc 890	Contraposition. Bidirectional version of con1dc 868. (Contributed by NM, 5-Aug-1993.)
|- (DECID ph -> (DECID ps -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) )
Theorem	con2biidc 891	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID ps -> ( ph <-> -. ps ) )   =>    |- (DECID ps -> ( ps <-> -. ph ) )
Theorem	con2biddc 892	A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
|- ( ph -> (DECID ch -> ( ps <-> -. ch ) ) )   =>    |- ( ph -> (DECID ch -> ( ch <-> -. ps ) ) )
Theorem	condandc 893	Proof by contradiction. This only holds for decidable propositions, as it is part of the family of theorems which assume -. ps, derive a contradiction, and therefore conclude ps. By contrast, assuming ps, deriving a contradiction, and therefore concluding -. ps, as in pm2.65 669, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
|- ( ( ph /\ -. ps ) -> ch )   &    |- ( ( ph /\ -. ps ) -> -. ch )   =>    |- (DECID ps -> ( ph -> ps ) )
Theorem	bijadc 894	Combine antecedents into a single biconditional. This inference is reminiscent of jadc 875. (Contributed by Jim Kingdon, 4-May-2018.)
|- ( ph -> ( ps -> ch ) )   &    |- ( -. ph -> ( -. ps -> ch ) )   =>    |- (DECID ps -> ( ( ph <-> ps ) -> ch ) )
Theorem	pm5.18dc 895	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 -. ( ph <-> -. ps ) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) )
Theorem	dfandc 896	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 646. (Contributed by Jim Kingdon, 30-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) )
Theorem	pm2.13dc 897	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 ph -> ( ph \/ -. -. -. ph ) )
Theorem	pm4.63dc 898	Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
|- (DECID ph -> (DECID ps -> ( -. ( ph -> -. ps ) <-> ( ph /\ ps ) ) ) )
Theorem	pm4.67dc 899	Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
|- (DECID ph -> (DECID ps -> ( -. ( -. ph -> -. ps ) <-> ( -. ph /\ ps ) ) ) )
Theorem	imanst 900	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 ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )