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	pm2.48 701	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( -. ( ph \/ ps ) -> ( ph \/ -. ps ) )
Theorem	pm2.49 702	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( -. ( ph \/ ps ) -> ( -. ph \/ -. ps ) )
Theorem	pm2.67 703	Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
|- ( ( ( ph \/ ps ) -> ps ) -> ( ph -> ps ) )
Theorem	biorf 704	A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
|- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
Theorem	biortn 705	A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
|- ( ph -> ( ps <-> ( -. ph \/ ps ) ) )
Theorem	biorfi 706	A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
|- -. ph   =>    |- ( ps <-> ( ps \/ ph ) )
Theorem	pm2.621 707	Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
|- ( ( ph -> ps ) -> ( ( ph \/ ps ) -> ps ) )
Theorem	pm2.62 708	Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
|- ( ( ph \/ ps ) -> ( ( ph -> ps ) -> ps ) )
Theorem	imorri 709	Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( -. ph \/ ps )   =>    |- ( ph -> ps )
Theorem	ioran 710	Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 851, anordc 908, or ianordc 843. 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	pm3.14 711	Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 843. The converse holds for decidable propositions, as seen at pm3.13dc 911. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )
Theorem	pm3.1 712	Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 908. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( ph /\ ps ) -> -. ( -. ph \/ -. ps ) )
Theorem	jao 713	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	pm1.2 714	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 715	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	pm4.25 716	Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph \/ ph ) )
Theorem	orim12i 717	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 718	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
|- ( ph -> ps )   =>    |- ( ( ph \/ ch ) -> ( ps \/ ch ) )
Theorem	orim2i 719	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
|- ( ph -> ps )   =>    |- ( ( ch \/ ph ) -> ( ch \/ ps ) )
Theorem	orbi2i 720	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 721	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 722	Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph \/ ch ) <-> ( ps \/ th ) )
Theorem	pm1.5 723	Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ps \/ ( ph \/ ch ) ) )
Theorem	or12 724	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 725	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	pm2.31 726	Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ( ph \/ ps ) \/ ch ) )
Theorem	pm2.32 727	Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ( ps \/ ch ) ) )
Theorem	or32 728	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 729	Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <-> ( ( ph \/ ch ) \/ ( ps \/ th ) ) )
Theorem	or42 730	Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <-> ( ( ph \/ ch ) \/ ( th \/ ps ) ) )
Theorem	orordi 731	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
|- ( ( ph \/ ( ps \/ ch ) ) <-> ( ( ph \/ ps ) \/ ( ph \/ ch ) ) )
Theorem	orordir 732	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
|- ( ( ( ph \/ ps ) \/ ch ) <-> ( ( ph \/ ch ) \/ ( ps \/ ch ) ) )
Theorem	pm2.3 733	Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ph \/ ( ch \/ ps ) ) )
Theorem	pm2.41 734	Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps \/ ( ph \/ ps ) ) -> ( ph \/ ps ) )
Theorem	pm2.42 735	Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( -. ph \/ ( ph -> ps ) ) -> ( ph -> ps ) )
Theorem	pm2.4 736	Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ph \/ ps ) ) -> ( ph \/ ps ) )
Theorem	pm4.44 737	Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph \/ ( ph /\ ps ) ) )
Theorem	mtord 738	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	pm4.45 739	Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph /\ ( ph \/ ps ) ) )
Theorem	pm3.48 740	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 741	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 742	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( ps \/ th ) -> ( ch \/ th ) ) )
Theorem	orim2d 743	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( th \/ ps ) -> ( th \/ ch ) ) )
Theorem	orim2 744	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) )
Theorem	orbi2d 745	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 746	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 747	Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
Theorem	orbi12d 748	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	pm5.61 749	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 750	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 751	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 752	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 753	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	pm4.77 754	Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) <-> ( ( ps \/ ch ) -> ph ) )
Theorem	pm2.63 755	Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( -. ph \/ ps ) -> ps ) )
Theorem	pm2.64 756	Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( ph \/ -. ps ) -> ph ) )
Theorem	pm5.53 757	Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ( ph \/ ps ) \/ ch ) -> th ) <-> ( ( ( ph -> th ) /\ ( ps -> th ) ) /\ ( ch -> th ) ) )
Theorem	pm2.38 758	Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ch \/ ph ) ) )
Theorem	pm2.36 759	Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ch \/ ph ) ) )
Theorem	pm2.37 760	Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
|- ( ( ps -> ch ) -> ( ( ps \/ ph ) -> ( ph \/ ch ) ) )
Theorem	pm2.73 761	Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph -> ps ) -> ( ( ( ph \/ ps ) \/ ch ) -> ( ps \/ ch ) ) )
Theorem	pm2.74 762	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	pm2.76 763	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	pm2.75 764	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	pm2.8 765	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	pm2.81 766	Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ( ch -> th ) ) -> ( ( ph \/ ps ) -> ( ( ph \/ ch ) -> ( ph \/ th ) ) ) )
Theorem	pm2.82 767	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 768	Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
|- -. ph   &    |- -. ps   =>    |- -. ( ph \/ ps )
Theorem	orabs 769	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 770	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 771	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 772	Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph /\ ps ) \/ ch ) <-> ( ( ph \/ ch ) /\ ( ps \/ ch ) ) )
Theorem	andi 773	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 774	Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )
Theorem	orddi 775	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 776	Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) /\ ( ch \/ th ) ) <-> ( ( ( ph /\ ch ) \/ ( ph /\ th ) ) \/ ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) )
Theorem	pm4.39 777	Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph <-> ch ) /\ ( ps <-> th ) ) -> ( ( ph \/ ps ) <-> ( ch \/ th ) ) )
Theorem	pm4.72 778	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	pm5.16 779	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 780	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 781	Extend wff definition to include stability.
wff STAB ph
Definition	df-stab 782	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 783	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 784	Every formula of the form -. ph is stable. Uses notnotnot 669. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- STAB -. ph
Theorem	imanst 785	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 ) ) )
1.2.8  Decidable propositions
Syntax	wdc 786	Extend wff definition to include decidability.
wff DECID ph
Definition	df-dc 787	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 793, exmiddc 788, peircedc 864, or notnotrdc 795, 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 788	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 789	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 790	A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
|- (DECID ph -> DECID -. ph )
Theorem	dcbii 791	The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- ( ph <-> ps )   =>    |- (DECID ph <-> DECID ps )
Theorem	dcbid 792	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 787), double negation elimination (notnotrdc 795), or contraposition (condc 793). 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 793	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 794	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 pm2.01 586 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
|- (DECID ph -> ( ( -. ph -> ph ) -> ph ) )
Theorem	notnotrdc 795	Double negation elimination for a decidable proposition. The converse, notnot 599, 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 796	Decidability implies stability. The converse is not necessarily true. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- (DECID ph -> STAB ph )
Theorem	con1dc 797	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 798	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 799	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 800	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 ) )