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	orim1d 701	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( ps \/ th ) -> ( ch \/ th ) ) )
Theorem	orim2d 702	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( th \/ ps ) -> ( th \/ ch ) ) )
Theorem	orim2 703	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) )
Theorem	orbi2d 704	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 705	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 706	Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
Theorem	orbi12d 707	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	pm5.61 708	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 709	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 710	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 711	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	pm4.77 712	Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) <-> ( ( ps \/ ch ) -> ph ) )
Theorem	pm2.63 713	Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( -. ph \/ ps ) -> ps ) )
Theorem	pm2.64 714	Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( ph \/ -. ps ) -> ph ) )
Theorem	pm5.53 715	Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ( ph \/ ps ) \/ ch ) -> th ) <-> ( ( ( ph -> th ) /\ ( ps -> th ) ) /\ ( ch -> th ) ) )
Theorem	pm2.38 716	Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ch \/ ph ) ) )
Theorem	pm2.36 717	Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ch \/ ph ) ) )
Theorem	pm2.37 718	Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ph \/ ch ) ) )
Theorem	pm2.73 719	Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph -> ps ) -> ( ( ( ph \/ ps ) \/ ch ) -> ( ps \/ ch ) ) )
Theorem	pm2.74 720	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	pm2.76 721	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	pm2.75 722	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	pm2.8 723	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	pm2.81 724	Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ( ch -> th ) ) -> ( ( ph \/ ps ) -> ( ( ph \/ ch ) -> ( ph \/ th ) ) ) )
Theorem	pm2.82 725	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 726	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
|- -. ph   &    |- -. ps   =>    |- -. ( ph \/ ps )
Theorem	orabs 727	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 728	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 729	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 730	Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph /\ ps ) \/ ch ) <-> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )
Theorem	andi 731	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 732	Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )
Theorem	orddi 733	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 734	Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) /\ ( ch \/ th ) ) <-> ( ( ( ph /\ ch ) \/ ( ph /\ th ) ) \/ ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) )
Theorem	pm4.39 735	Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) -> ( ( ph \/ ps ) <-> ( ch \/ th ) ) )
Theorem	pm4.72 736	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	pm5.16 737	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 738	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 739	Extend wff definition to include stability.
wff STAB ph
Definition	df-stab 740	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 741	Every formula of the form -. ph is stable. Uses notnotnot 628. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- STAB -. ph
1.2.8  Decidable propositions
Syntax	wdc 742	Extend wff definition to include decidability.
wff DECID ph
Definition	df-dc 743	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 749, exmiddc 744, peircedc 820, or notnotrdc 751, 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 744	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 745	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 746	A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
|- (DECID ph -> DECID -. ph )
Theorem	dcbii 747	The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- ( ph <-> ps )   =>    |- (DECID ph <-> DECID ps )
Theorem	dcbid 748	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 743), double negation elimination (notnotrdc 751), or contraposition (condc 749). 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 749	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 750	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	notnotrdc 751	Double negation elimination for a decidable proposition. The converse, notnot 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 752	Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- (DECID ph -> STAB ph )
Theorem	con1dc 753	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 754	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 755	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 756	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 757	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 758	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 759	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 760	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 761	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 762	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 763	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 764	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 765	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 766	Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 559, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
|- (DECID ph -> ( ph <-> -. -. ph ) )
Theorem	con1biimdc 767	Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
|- (DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps <-> ph ) ) )
Theorem	con1bidc 768	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) ) )
Theorem	con2bidc 769	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) )
Theorem	con1biddc 770	A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
|- ( ph -> (DECID ps -> ( -. ps <-> ch ) ) )   =>    |- ( ph -> (DECID ps -> ( -. ch <-> ps ) ) )
Theorem	con1biidc 771	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID ph -> ( -. ph <-> ps ) )   =>    |- (DECID ph -> ( -. ps <-> ph ) )
Theorem	con1bdc 772	Contraposition. Bidirectional version of con1dc 753. (Contributed by NM, 5-Aug-1993.)
|- (DECID ph -> (DECID ps -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) )
Theorem	con2biidc 773	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID ps -> ( ph <-> -. ps ) )   =>    |- (DECID ps -> ( ps <-> -. ph ) )
Theorem	con2biddc 774	A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
|- ( ph -> (DECID ch -> ( ps <-> -. ch ) ) )   =>    |- ( ph -> (DECID ch -> ( ch <-> -. ps ) ) )
Theorem	condandc 775	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 pm2.65 585, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
|- ( ( ph /\ -. ps ) -> ch )   &    |- ( ( ph /\ -. ps ) -> -. ch )   =>    |- (DECID ps -> ( ph -> ps ) )
Theorem	bijadc 776	Combine antecedents into a single biconditional. This inference is reminiscent of jadc 760. (Contributed by Jim Kingdon, 4-May-2018.)
|- ( ph -> ( ps -> ch ) )   &    |- ( -. ph -> ( -. ps -> ch ) )   =>    |- (DECID ps -> ( ( ph <-> ps ) -> ch ) )
Theorem	pm5.18dc 777	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 778	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 566. (Contributed by Jim Kingdon, 30-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) )
Theorem	pm2.13dc 779	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 780	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 781	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 782	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 845. (Contributed by Jim Kingdon, 24-Dec-2017.)
|- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )
Theorem	pm4.65r 783	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 784	An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
Theorem	imanim 785	Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 786. (Contributed by Jim Kingdon, 24-Dec-2017.)
|- ( ( ph -> ps ) -> -. ( ph /\ -. ps ) )
Theorem	imandc 786	Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 785, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
|- (DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )
Theorem	pm4.14dc 787	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 788	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	pm4.15 789	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 790	Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 641, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
|- (DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) )
Theorem	dfordc 791	Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 641, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)
|- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )
Theorem	pm2.25dc 792	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 793	Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 667 and one half of dfor2dc 794. (Contributed by Jim Kingdon, 27-Mar-2018.)
|- (DECID ph -> ( ( ( ph -> ps ) -> ps ) -> ( ph \/ ps ) ) )
Theorem	dfor2dc 794	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 795	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 796	Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 797, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
|- (DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) )
Theorem	imorr 797	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 796. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ( ( -. ph \/ ps ) -> ( ph -> ps ) )
Theorem	pm4.62dc 798	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 799	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, pm3.14 670, 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 800	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 ) )