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	con1bidc 801	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) ) )
Theorem	con2bidc 802	Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) )
Theorem	con1biddc 803	A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
|- ( ph -> (DECID ps -> ( -. ps <-> ch ) ) )   =>    |- ( ph -> (DECID ps -> ( -. ch <-> ps ) ) )
Theorem	con1biidc 804	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID ph -> ( -. ph <-> ps ) )   =>    |- (DECID ph -> ( -. ps <-> ph ) )
Theorem	con1bdc 805	Contraposition. Bidirectional version of con1dc 786. (Contributed by NM, 5-Aug-1993.)
|- (DECID ph -> (DECID ps -> ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) ) )
Theorem	con2biidc 806	A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID ps -> ( ph <-> -. ps ) )   =>    |- (DECID ps -> ( ps <-> -. ph ) )
Theorem	con2biddc 807	A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
|- ( ph -> (DECID ch -> ( ps <-> -. ch ) ) )   =>    |- ( ph -> (DECID ch -> ( ch <-> -. ps ) ) )
Theorem	condandc 808	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 pm2.65 617, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
|- ( ( ph /\ -. ps ) -> ch )   &    |- ( ( ph /\ -. ps ) -> -. ch )   =>    |- (DECID ps -> ( ph -> ps ) )
Theorem	bijadc 809	Combine antecedents into a single biconditional. This inference is reminiscent of jadc 793. (Contributed by Jim Kingdon, 4-May-2018.)
|- ( ph -> ( ps -> ch ) )   &    |- ( -. ph -> ( -. ps -> ch ) )   =>    |- (DECID ps -> ( ( ph <-> ps ) -> ch ) )
Theorem	pm5.18dc 810	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 811	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 598. (Contributed by Jim Kingdon, 30-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) )
Theorem	pm2.13dc 812	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 813	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 814	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 815	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 878. (Contributed by Jim Kingdon, 24-Dec-2017.)
|- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )
Theorem	pm4.65r 816	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 817	An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
Theorem	imanim 818	Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 819. (Contributed by Jim Kingdon, 24-Dec-2017.)
|- ( ( ph -> ps ) -> -. ( ph /\ -. ps ) )
Theorem	imandc 819	Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 818, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
|- (DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )
Theorem	pm4.14dc 820	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	pm3.37 821	Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph /\ ps ) -> ch ) -> ( ( ph /\ -. ch ) -> -. ps ) )
Theorem	pm4.15 822	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 823	Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 673, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
|- (DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) )
Theorem	dfordc 824	Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 673, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)
|- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )
Theorem	pm2.25dc 825	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 826	Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 699 and one half of dfor2dc 827. (Contributed by Jim Kingdon, 27-Mar-2018.)
|- (DECID ph -> ( ( ( ph -> ps ) -> ps ) -> ( ph \/ ps ) ) )
Theorem	dfor2dc 827	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 828	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 829	Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 830, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
|- (DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) )
Theorem	imorr 830	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 829. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ( ( -. ph \/ ps ) -> ( ph -> ps ) )
Theorem	pm4.62dc 831	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 832	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, pm3.14 702, 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 833	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 834	Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 673, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)
|- (DECID ph -> ( ( -. ph -> ps ) <-> ( ph \/ ps ) ) )
Theorem	pm4.66dc 835	Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability condition. (Contributed by Jim Kingdon, 2-May-2018.)
|- (DECID ph -> ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) ) )
Theorem	pm4.52im 836	One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
|- ( ( ph /\ -. ps ) -> -. ( -. ph \/ ps ) )
Theorem	pm4.53r 837	One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
|- ( ( -. ph \/ ps ) -> -. ( ph /\ -. ps ) )
Theorem	pm4.54dc 838	Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)
|- (DECID ph -> (DECID ps -> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) )
Theorem	pm4.56 839	Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
|- ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) )
Theorem	oranim 840	Disjunction in terms of conjunction (DeMorgan's law). One direction of Theorem *4.57 of [WhiteheadRussell] p. 120. The converse does not hold intuitionistically but does hold in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
|- ( ( ph \/ ps ) -> -. ( -. ph /\ -. ps ) )
Theorem	pm4.78i 841	Implication distributes over disjunction. One direction of Theorem *4.78 of [WhiteheadRussell] p. 121. The converse holds in classical logic. (Contributed by Jim Kingdon, 15-Jan-2018.)
|- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) -> ( ph -> ( ps \/ ch ) ) )
Theorem	pm4.79dc 842	Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)
|- (DECID ph -> (DECID ps -> ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <-> ( ( ps /\ ch ) -> ph ) ) ) )
Theorem	pm5.17dc 843	Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)
|- (DECID ps -> ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) )
Theorem	pm2.85dc 844	Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.)
|- (DECID ph -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) -> ( ph \/ ( ps -> ch ) ) ) )
Theorem	orimdidc 845	Disjunction distributes over implication. The forward direction, pm2.76 754, is valid intuitionistically. The reverse direction holds if ph is decidable, as can be seen at pm2.85dc 844. (Contributed by Jim Kingdon, 1-Apr-2018.)
|- (DECID ph -> ( ( ph \/ ( ps -> ch ) ) <-> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) )
Theorem	pm2.26dc 846	Decidable proposition version of theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.)
|- (DECID ph -> ( -. ph \/ ( ( ph -> ps ) -> ps ) ) )
Theorem	pm4.81dc 847	Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 655 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)
|- (DECID ph -> ( ( -. ph -> ph ) <-> ph ) )
Theorem	pm5.11dc 848	A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) )
Theorem	pm5.12dc 849	Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
|- (DECID ps -> ( ( ph -> ps ) \/ ( ph -> -. ps ) ) )
Theorem	pm5.14dc 850	A decidable proposition is implied by or implies other propositions. Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
|- (DECID ps -> ( ( ph -> ps ) \/ ( ps -> ch ) ) )
Theorem	pm5.13dc 851	An implication holds in at least one direction, where one proposition is decidable. Based on theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
|- (DECID ps -> ( ( ph -> ps ) \/ ( ps -> ph ) ) )
Theorem	pm5.55dc 852	A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
|- (DECID ph -> ( ( ( ph \/ ps ) <-> ph ) \/ ( ( ph \/ ps ) <-> ps ) ) )
Theorem	peircedc 853	Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 777, condc 782, or notnotrdc 784 in the sense of expressing the "difference" between an intuitionistic system of propositional calculus and a classical system. In intuitionistic logic, it only holds for decidable propositions. (Contributed by Jim Kingdon, 3-Jul-2018.)
|- (DECID ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
Theorem	looinvdc 854	The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 827, we can see that this expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability condition). (Contributed by NM, 12-Aug-2004.)
|- (DECID ph -> ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) )
1.2.10  Testable propositions
Theorem	dftest 855	A proposition is testable iff its negative or double-negative is true. See Chapter 2 [Moschovakis] p. 2. Our notation for testability is DECID -. before the formula in question. For example, DECID -. x = y corresponds to "x = y is testable". (Contributed by David A. Wheeler, 13-Aug-2018.)
|- (DECID -. ph <-> ( -. ph \/ -. -. ph ) )
Theorem	testbitestn 856	A proposition is testable iff its negation is testable. See also dcn 779 (which could be read as "Decidability implies testability"). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- (DECID -. ph <-> DECID -. -. ph )
Theorem	stabtestimpdc 857	"Stable and testable" is equivalent to decidable. (Contributed by David A. Wheeler, 13-Aug-2018.)
|- ( (STAB ph /\ DECID -. ph ) <-> DECID ph )
1.2.11  Miscellaneous theorems of propositional calculus
Theorem	pm5.21nd 858	Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
|- ( ( ph /\ ps ) -> th )   &    |- ( ( ph /\ ch ) -> th )   &    |- ( th -> ( ps <-> ch ) )   =>    |- ( ph -> ( ps <-> ch ) )
Theorem	pm5.35 859	Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) -> ( ph -> ( ps <-> ch ) ) )
Theorem	pm5.54dc 860	A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
|- (DECID ph -> ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) ) )
Theorem	baib 861	Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ps -> ( ph <-> ch ) )
Theorem	baibr 862	Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ps -> ( ch <-> ph ) )
Theorem	rbaib 863	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ch -> ( ph <-> ps ) )
Theorem	rbaibr 864	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ch -> ( ps <-> ph ) )
Theorem	baibd 865	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
|- ( ph -> ( ps <-> ( ch /\ th ) ) )   =>    |- ( ( ph /\ ch ) -> ( ps <-> th ) )
Theorem	rbaibd 866	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
|- ( ph -> ( ps <-> ( ch /\ th ) ) )   =>    |- ( ( ph /\ th ) -> ( ps <-> ch ) )
Theorem	pm5.44 867	Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph -> ps ) -> ( ( ph -> ch ) <-> ( ph -> ( ps /\ ch ) ) ) )
Theorem	pm5.6dc 868	Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 869). (Contributed by Jim Kingdon, 2-Apr-2018.)
|- (DECID ps -> ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( ps \/ ch ) ) ) )
Theorem	pm5.6r 869	Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If ps is decidable, the converse also holds (see pm5.6dc 868). (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( ( ph -> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) -> ch ) )
Theorem	orcanai 870	Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
|- ( ph -> ( ps \/ ch ) )   =>    |- ( ( ph /\ -. ps ) -> ch )
Theorem	intnan 871	Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
|- -. ph   =>    |- -. ( ps /\ ph )
Theorem	intnanr 872	Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
|- -. ph   =>    |- -. ( ph /\ ps )
Theorem	intnand 873	Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
|- ( ph -> -. ps )   =>    |- ( ph -> -. ( ch /\ ps ) )
Theorem	intnanrd 874	Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
|- ( ph -> -. ps )   =>    |- ( ph -> -. ( ps /\ ch ) )
Theorem	dcan 875	A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
|- (DECID ph -> (DECID ps -> DECID ( ph /\ ps ) ) )
Theorem	dcor 876	A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
|- (DECID ph -> (DECID ps -> DECID ( ph \/ ps ) ) )
Theorem	dcbi 877	An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
|- (DECID ph -> (DECID ps -> DECID ( ph <-> ps ) ) )
Theorem	annimdc 878	Express conjunction in terms of implication. The forward direction, annimim 815, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) )
Theorem	pm4.55dc 879	Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
|- (DECID ph -> (DECID ps -> ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) ) )
Theorem	orandc 880	Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)
|- ( (DECID ph /\ DECID ps ) -> ( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) )
Theorem	mpbiran 881	Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)
|- ps   &    |- ( ph <-> ( ps /\ ch ) )   =>    |- ( ph <-> ch )
Theorem	mpbiran2 882	Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)
|- ch   &    |- ( ph <-> ( ps /\ ch ) )   =>    |- ( ph <-> ps )
Theorem	mpbir2an 883	Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.) (Revised by NM, 9-Jan-2015.)
|- ps   &    |- ch   &    |- ( ph <-> ( ps /\ ch ) )   =>    |- ph
Theorem	mpbi2and 884	Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ph -> ( ( ps /\ ch ) <-> th ) )   =>    |- ( ph -> th )
Theorem	mpbir2and 885	Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
|- ( ph -> ch )   &    |- ( ph -> th )   &    |- ( ph -> ( ps <-> ( ch /\ th ) ) )   =>    |- ( ph -> ps )
Theorem	pm5.62dc 886	Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
|- (DECID ps -> ( ( ( ph /\ ps ) \/ -. ps ) <-> ( ph \/ -. ps ) ) )
Theorem	pm5.63dc 887	Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
|- (DECID ph -> ( ( ph \/ ps ) <-> ( ph \/ ( -. ph /\ ps ) ) ) )
Theorem	bianfi 888	A wff conjoined with falsehood is false. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
|- -. ph   =>    |- ( ph <-> ( ps /\ ph ) )
Theorem	bianfd 889	A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.)
|- ( ph -> -. ps )   =>    |- ( ph -> ( ps <-> ( ps /\ ch ) ) )
Theorem	pm4.43 890	Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
|- ( ph <-> ( ( ph \/ ps ) /\ ( ph \/ -. ps ) ) )
Theorem	pm4.82 891	Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph -> ps ) /\ ( ph -> -. ps ) ) <-> -. ph )
Theorem	pm4.83dc 892	Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 795, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
|- (DECID ph -> ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) <-> ps ) )
Theorem	biantr 893	A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
|- ( ( ( ph <-> ps ) /\ ( ch <-> ps ) ) -> ( ph <-> ch ) )
Theorem	orbididc 894	Disjunction distributes over the biconditional, for a decidable proposition. Based on an axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by Jim Kingdon, 2-Apr-2018.)
|- (DECID ph -> ( ( ph \/ ( ps <-> ch ) ) <-> ( ( ph \/ ps ) <-> ( ph \/ ch ) ) ) )
Theorem	pm5.7dc 895	Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 894. (Contributed by Jim Kingdon, 2-Apr-2018.)
|- (DECID ch -> ( ( ( ph \/ ch ) <-> ( ps \/ ch ) ) <-> ( ch \/ ( ph <-> ps ) ) ) )
Theorem	bigolden 896	Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
|- ( ( ( ph /\ ps ) <-> ph ) <-> ( ps <-> ( ph \/ ps ) ) )
Theorem	anordc 897	Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 703, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) )
Theorem	pm3.11dc 898	Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 703, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) ) )
Theorem	pm3.12dc 899	Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( ( -. ph \/ -. ps ) \/ ( ph /\ ps ) ) ) )
Theorem	pm3.13dc 900	Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 702, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
|- (DECID ph -> (DECID ps -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) ) )