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	ioran 701	Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 840, anordc 897, or ianordc 832. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) )
Theorem	pm3.14 702	Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 832. The converse holds for decidable propositions, as seen at pm3.13dc 900. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )
Theorem	pm3.1 703	Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 897. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( ph /\ ps ) -> -. ( -. ph \/ -. ps ) )
Theorem	jao 704	Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.)
|- ( ( ph -> ps ) -> ( ( ch -> ps ) -> ( ( ph \/ ch ) -> ps ) ) )
Theorem	pm1.2 705	Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
|- ( ( ph \/ ph ) -> ph )
Theorem	oridm 706	Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.)
|- ( ( ph \/ ph ) <-> ph )
Theorem	pm4.25 707	Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph \/ ph ) )
Theorem	orim12i 708	Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
|- ( ph -> ps )   &    |- ( ch -> th )   =>    |- ( ( ph \/ ch ) -> ( ps \/ th ) )
Theorem	orim1i 709	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
|- ( ph -> ps )   =>    |- ( ( ph \/ ch ) -> ( ps \/ ch ) )
Theorem	orim2i 710	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
|- ( ph -> ps )   =>    |- ( ( ch \/ ph ) -> ( ch \/ ps ) )
Theorem	orbi2i 711	Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.)
|- ( ph <-> ps )   =>    |- ( ( ch \/ ph ) <-> ( ch \/ ps ) )
Theorem	orbi1i 712	Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- ( ( ph \/ ch ) <-> ( ps \/ ch ) )
Theorem	orbi12i 713	Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph \/ ch ) <-> ( ps \/ th ) )
Theorem	pm1.5 714	Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ps \/ ( ph \/ ch ) ) )
Theorem	or12 715	Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
|- ( ( ph \/ ( ps \/ ch ) ) <-> ( ps \/ ( ph \/ ch ) ) )
Theorem	orass 716	Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) )
Theorem	pm2.31 717	Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ( ph \/ ps ) \/ ch ) )
Theorem	pm2.32 718	Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ( ps \/ ch ) ) )
Theorem	or32 719	A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( ( ph \/ ps ) \/ ch ) <-> ( ( ph \/ ch ) \/ ps ) )
Theorem	or4 720	Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <-> ( ( ph \/ ch ) \/ ( ps \/ th ) ) )
Theorem	or42 721	Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <-> ( ( ph \/ ch ) \/ ( th \/ ps ) ) )
Theorem	orordi 722	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
|- ( ( ph \/ ( ps \/ ch ) ) <-> ( ( ph \/ ps ) \/ ( ph \/ ch ) ) )
Theorem	orordir 723	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
|- ( ( ( ph \/ ps ) \/ ch ) <-> ( ( ph \/ ch ) \/ ( ps \/ ch ) ) )
Theorem	pm2.3 724	Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ph \/ ( ch \/ ps ) ) )
Theorem	pm2.41 725	Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps \/ ( ph \/ ps ) ) -> ( ph \/ ps ) )
Theorem	pm2.42 726	Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( -. ph \/ ( ph -> ps ) ) -> ( ph -> ps ) )
Theorem	pm2.4 727	Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ph \/ ps ) ) -> ( ph \/ ps ) )
Theorem	pm4.44 728	Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph \/ ( ph /\ ps ) ) )
Theorem	mtord 729	A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ph -> -. ch )   &    |- ( ph -> -. th )   &    |- ( ph -> ( ps -> ( ch \/ th ) ) )   =>    |- ( ph -> -. ps )
Theorem	pm4.45 730	Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph /\ ( ph \/ ps ) ) )
Theorem	pm3.48 731	Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.)
|- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ( ph \/ ch ) -> ( ps \/ th ) ) )
Theorem	orim12d 732	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th -> ta ) )   =>    |- ( ph -> ( ( ps \/ th ) -> ( ch \/ ta ) ) )
Theorem	orim1d 733	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( ps \/ th ) -> ( ch \/ th ) ) )
Theorem	orim2d 734	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( th \/ ps ) -> ( th \/ ch ) ) )
Theorem	orim2 735	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) )
Theorem	orbi2d 736	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 737	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 738	Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
Theorem	orbi12d 739	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	pm5.61 740	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 741	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 742	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 743	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 744	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	pm4.77 745	Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) <-> ( ( ps \/ ch ) -> ph ) )
Theorem	pm2.63 746	Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( -. ph \/ ps ) -> ps ) )
Theorem	pm2.64 747	Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( ph \/ -. ps ) -> ph ) )
Theorem	pm5.53 748	Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ( ph \/ ps ) \/ ch ) -> th ) <-> ( ( ( ph -> th ) /\ ( ps -> th ) ) /\ ( ch -> th ) ) )
Theorem	pm2.38 749	Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ch \/ ph ) ) )
Theorem	pm2.36 750	Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ch \/ ph ) ) )
Theorem	pm2.37 751	Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ph \/ ch ) ) )
Theorem	pm2.73 752	Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph -> ps ) -> ( ( ( ph \/ ps ) \/ ch ) -> ( ps \/ ch ) ) )
Theorem	pm2.74 753	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	pm2.76 754	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	pm2.75 755	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	pm2.8 756	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	pm2.81 757	Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ( ch -> th ) ) -> ( ( ph \/ ps ) -> ( ( ph \/ ch ) -> ( ph \/ th ) ) ) )
Theorem	pm2.82 758	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 759	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
|- -. ph   &    |- -. ps   =>    |- -. ( ph \/ ps )
Theorem	orabs 760	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 761	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 762	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 763	Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph /\ ps ) \/ ch ) <-> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )
Theorem	andi 764	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 765	Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )
Theorem	orddi 766	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 767	Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) /\ ( ch \/ th ) ) <-> ( ( ( ph /\ ch ) \/ ( ph /\ th ) ) \/ ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) )
Theorem	pm4.39 768	Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) -> ( ( ph \/ ps ) <-> ( ch \/ th ) ) )
Theorem	pm4.72 769	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	pm5.16 770	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 771	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 772	Extend wff definition to include stability.
wff STAB ph
Definition	df-stab 773	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 774	Every formula of the form -. ph is stable. Uses notnotnot 660. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- STAB -. ph
1.2.8  Decidable propositions
Syntax	wdc 775	Extend wff definition to include decidability.
wff DECID ph
Definition	df-dc 776	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 782, exmiddc 777, peircedc 853, or notnotrdc 784, 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 777	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 778	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 779	A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
|- (DECID ph -> DECID -. ph )
Theorem	dcbii 780	The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- ( ph <-> ps )   =>    |- (DECID ph <-> DECID ps )
Theorem	dcbid 781	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 776), double negation elimination (notnotrdc 784), or contraposition (condc 782). 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 782	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 783	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 pm2.01 578 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
|- (DECID ph -> ( ( -. ph -> ph ) -> ph ) )
Theorem	notnotrdc 784	Double negation elimination for a decidable proposition. The converse, notnot 591, 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 785	Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- (DECID ph -> STAB ph )
Theorem	con1dc 786	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 787	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 788	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 789	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 790	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 791	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 792	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 793	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 794	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 795	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 796	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 797	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 798	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 799	Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 591, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
|- (DECID ph -> ( ph <-> -. -. ph ) )
Theorem	con1biimdc 800	Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
|- (DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps <-> ph ) ) )