P. 8 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	orim2d 701	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th \/ ps) -> (th \/ ch)))
Theorem	orim2 702	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
|- ((ps -> ch) -> ((ph \/ ps) -> (ph \/ ch)))
Theorem	orbi2d 703	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 704	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 705	Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ((ph <-> ps) -> ((ph \/ ch) <-> (ps \/ ch)))
Theorem	orbi12d 706	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	pm5.61 707	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 708	Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
|- ((ph /\ ps) -> ch)   &   |- ((th /\ ps) -> ch)   =>   |- (((ph \/ th) /\ ps) -> ch)
Theorem	jaodan 709	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 710	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	pm4.77 711	Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
|- (((ps -> ph) /\ (ch -> ph)) <-> ((ps \/ ch) -> ph))
Theorem	pm2.63 712	Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ((ph \/ ps) -> ((-. ph \/ ps) -> ps))
Theorem	pm2.64 713	Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ((ph \/ ps) -> ((ph \/ -. ps) -> ph))
Theorem	pm5.53 714	Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
|- ((((ph \/ ps) \/ ch) -> th) <-> (((ph -> th) /\ (ps -> th)) /\ (ch -> th)))
Theorem	pm2.38 715	Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ((ps -> ch) -> ((ps \/ ph) -> (ch \/ ph)))
Theorem	pm2.36 716	Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ((ps -> ch) -> ((ph \/ ps) -> (ch \/ ph)))
Theorem	pm2.37 717	Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ((ps -> ch) -> ((ps \/ ph) -> (ph \/ ch)))
Theorem	pm2.73 718	Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ((ph -> ps) -> (((ph \/ ps) \/ ch) -> (ps \/ ch)))
Theorem	pm2.74 719	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	pm2.76 720	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	pm2.75 721	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	pm2.8 722	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	pm2.81 723	Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ((ps -> (ch -> th)) -> ((ph \/ ps) -> ((ph \/ ch) -> (ph \/ th))))
Theorem	pm2.82 724	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 725	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
|- -. ph   &   |- -. ps   =>   |- -. (ph \/ ps)
Theorem	orabs 726	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 727	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 728	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 729	Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
|- (((ph /\ ps) \/ ch) <-> ((ph \/ ch) /\ (ps \/ ch)))
Theorem	andi 730	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 731	Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- (((ph \/ ps) /\ ch) <-> ((ph /\ ch) \/ (ps /\ ch)))
Theorem	orddi 732	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 733	Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- (((ph \/ ps) /\ (ch \/ th)) <-> (((ph /\ ch) \/ (ph /\ th)) \/ ((ps /\ ch) \/ (ps /\ th))))
Theorem	pm4.39 734	Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- (((ph <-> ch) /\ (ps <-> th)) -> ((ph \/ ps) <-> (ch \/ th)))
Theorem	pm4.72 735	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	pm5.16 736	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 737	A wff is disjoined with truth is true. (Contributed by NM, 23-May-1999.)
|- (ph -> (ph <-> (ph \/ ps)))
1.2.7  Stable propositions
Syntax	wstab 738	Extend wff definition to include stability.
wff STAB ph
Definition	df-stab 739	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	stabnot 740	Every formula of the form -. ph is stable. Uses notnotnot 627. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- STAB -. ph
1.2.8  Decidable propositions
Syntax	wdc 741	Extend wff definition to include decidability.
wff DECID ph
Definition	df-dc 742	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 748, exmiddc 743, peircedc 819, or notnot2dc 750, 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 743	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 744	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	dcn 745	A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
|- (DECID ph -> DECID -. ph)
Theorem	dcbii 746	The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- (ph <-> ps)   =>   |- (DECID ph <-> DECID ps)
Theorem	dcbid 747	The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
|- (ph -> (ps <-> ch))   =>   |- (ph -> (DECID ps <-> DECID ch))
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 742), double negation elimination (notnotdc 765), or contraposition (condc 748). 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 748	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 ph -> ((-. ph -> -. ps) -> (ps -> ph)))
Theorem	pm2.18dc 749	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 pm2.01 546 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
|- (DECID ph -> ((-. ph -> ph) -> ph))
Theorem	notnot2dc 750	Double negation elimination for a decidable proposition. The converse, notnot1 559, 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	dcimpstab 751	Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- (DECID ph -> STAB ph)
Theorem	con1dc 752	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 753	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 754	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 755	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 756	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 757	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 758	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 759	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 760	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 761	Case elimination for a decidable proposition. Based on theorem *2.61 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.)
|- (DECID ph -> ((ph -> ps) -> ((-. ph -> ps) -> ps)))
Theorem	pm2.5dc 762	Negating an implication for a decidable antecedent. Based on theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.)
|- (DECID ph -> (-. (ph -> ps) -> (-. ph -> ps)))
Theorem	pm2.521dc 763	Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. (Contributed by Jim Kingdon, 5-May-2018.)
|- (DECID ph -> (-. (ph -> ps) -> (ps -> ph)))
Theorem	con34bdc 764	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	notnotdc 765	Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot1 559, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
|- (DECID ph -> (ph <-> -. -. ph))
Theorem	con1biimdc 766	Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
|- (DECID ph -> ((-. ph <-> ps) -> (-. ps <-> ph)))
Theorem	con1bidc 767	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ((-. ph <-> ps) <-> (-. ps <-> ph))))
Theorem	con2bidc 768	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ((ph <-> -. ps) <-> (ps <-> -. ph))))
Theorem	con1biddc 769	A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
|- (ph -> (DECID ps -> (-. ps <-> ch)))   =>   |- (ph -> (DECID ps -> (-. ch <-> ps)))
Theorem	con1biidc 770	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID ph -> (-. ph <-> ps))   =>   |- (DECID ph -> (-. ps <-> ph))
Theorem	con1bdc 771	Contraposition. Bidirectional version of con1dc 752. (Contributed by NM, 5-Aug-1993.)
|- (DECID ph -> (DECID ps -> ((-. ph -> ps) <-> (-. ps -> ph))))
Theorem	con2biidc 772	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID ps -> (ph <-> -. ps))   =>   |- (DECID ps -> (ps <-> -. ph))
Theorem	con2biddc 773	A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
|- (ph -> (DECID ch -> (ps <-> -. ch)))   =>   |- (ph -> (DECID ch -> (ch <-> -. ps)))
Theorem	condandc 774	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 pm2.65 584, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
|- ((ph /\ -. ps) -> ch)   &   |- ((ph /\ -. ps) -> -. ch)   =>   |- (DECID ps -> (ph -> ps))
Theorem	bijadc 775	Combine antecedents into a single biconditional. This inference is reminiscent of jadc 759. (Contributed by Jim Kingdon, 4-May-2018.)
|- (ph -> (ps -> ch))   &   |- (-. ph -> (-. ps -> ch))   =>   |- (DECID ps -> ((ph <-> ps) -> ch))
Theorem	pm5.18dc 776	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 777	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 565. (Contributed by Jim Kingdon, 30-Apr-2018.)
|- (DECID ph -> (DECID ps -> ((ph /\ ps) <-> -. (ph -> -. ps))))
Theorem	pm2.13dc 778	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 779	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 780	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	annimim 781	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 844. (Contributed by Jim Kingdon, 24-Dec-2017.)
|- ((ph /\ -. ps) -> -. (ph -> ps))
Theorem	pm4.65r 782	One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- ((-. ph /\ -. ps) -> -. (-. ph -> ps))
Theorem	dcim 783	An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- (DECID ph -> (DECID ps -> DECID (ph -> ps)))
Theorem	imanim 784	Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 785. (Contributed by Jim Kingdon, 24-Dec-2017.)
|- ((ph -> ps) -> -. (ph /\ -. ps))
Theorem	imandc 785	Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 784, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
|- (DECID ps -> ((ph -> ps) <-> -. (ph /\ -. ps)))
Theorem	pm4.14dc 786	Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
|- (DECID ch -> (((ph /\ ps) -> ch) <-> ((ph /\ -. ch) -> -. ps)))
Theorem	pm3.37dc 787	Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
|- (DECID ch -> (((ph /\ ps) -> ch) -> ((ph /\ -. ch) -> -. ps)))
Theorem	pm4.15 788	Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
|- (((ph /\ ps) -> -. ch) <-> ((ps /\ ch) -> -. ph))
Theorem	pm2.54dc 789	Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 640, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
|- (DECID ph -> ((-. ph -> ps) -> (ph \/ ps)))
Theorem	dfordc 790	Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 640, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)
|- (DECID ph -> ((ph \/ ps) <-> (-. ph -> ps)))
Theorem	pm2.25dc 791	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 ph -> (ph \/ ((ph \/ ps) -> ps)))
Theorem	pm2.68dc 792	Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 666 and one half of dfor2dc 793. (Contributed by Jim Kingdon, 27-Mar-2018.)
|- (DECID ph -> (((ph -> ps) -> ps) -> (ph \/ ps)))
Theorem	dfor2dc 793	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 ph -> ((ph \/ ps) <-> ((ph -> ps) -> ps)))
Theorem	imimorbdc 794	Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)
|- (DECID ps -> (((ps -> ch) -> (ph -> ch)) <-> (ph -> (ps \/ ch))))
Theorem	imordc 795	Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 796, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
|- (DECID ph -> ((ph -> ps) <-> (-. ph \/ ps)))
Theorem	imorr 796	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 795. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ((-. ph \/ ps) -> (ph -> ps))
Theorem	pm4.62dc 797	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 ph -> ((ph -> -. ps) <-> (-. ph \/ -. ps)))
Theorem	ianordc 798	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, pm3.14 669, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
|- (DECID ph -> (-. (ph /\ ps) <-> (-. ph \/ -. ps)))
Theorem	oibabs 799	Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
|- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))
Theorem	pm4.64dc 800	Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 640, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)
|- (DECID ph -> ((-. ph -> ps) <-> (ph \/ ps)))