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

Theorem	pm4.15 701	Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
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Theorem	pm5.21 702	Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	pm5.21im 703	Two propositions are equivalent if they are both false. Closed form of 2false 708. Equivalent to a biimpr 130-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	nbn2 704	The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	bibif 705	Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)


Theorem	nbn 706	The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)


Theorem	nbn3 707	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)


Theorem	2false 708	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	2falsed 709	Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)


Theorem	pm5.21ni 710	Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)


Theorem	pm5.21nii 711	Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm5.21ndd 712	Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm5.19 713	Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm4.8 714	Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 915 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)


1.2.6 Logical disjunction

Syntax	wo 715	Extend wff definition to include disjunction ('or').
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Axiom	ax-io 716	Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 717 instead. (New usage is discouraged.)
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Theorem	jaob 717	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 716. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	olc 718	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
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Theorem	orc 719	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
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Theorem	pm2.67-2 720	Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
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Theorem	oibabs 721	Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
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Theorem	pm3.44 722	Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
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Theorem	jaoi 723	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
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Theorem	jaod 724	Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.)
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Theorem	mpjaod 725	Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
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Theorem	jaao 726	Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
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Theorem	jaoa 727	Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
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Theorem	imorr 728	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 904. (Contributed by Jim Kingdon, 21-Jul-2018.)
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Theorem	pm2.53 729	Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 898). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
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Theorem	ori 730	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	ord 731	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	orel1 732	Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 21-Jul-2012.)
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Theorem	orel2 733	Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Apr-2013.)
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Theorem	pm1.4 734	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
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Theorem	orcom 735	Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)
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Theorem	orcomd 736	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
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Theorem	orcoms 737	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
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Theorem	orci 738	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	olci 739	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	orcd 740	Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
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Theorem	olcd 741	Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
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Theorem	orcs 742	Deduction eliminating disjunct. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 14) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)
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Theorem	olcs 743	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
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Theorem	pm2.07 744	Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.45 745	Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.46 746	Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.47 747	Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.48 748	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.49 749	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.67 750	Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
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Theorem	biorf 751	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.)
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Theorem	biortn 752	A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
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Theorem	biorfi 753	A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
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Theorem	pm2.621 754	Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
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Theorem	pm2.62 755	Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
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Theorem	imorri 756	Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	pm4.52im 757	One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
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Theorem	pm4.53r 758	One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
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Theorem	ioran 759	Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 788, anordc 964, or ianordc 906. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	pm3.14 760	Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 906. The converse holds for decidable propositions, as seen at pm3.13dc 967. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	pm3.1 761	Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 964. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	jao 762	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.)
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Theorem	pm1.2 763	Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
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Theorem	oridm 764	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.)
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Theorem	pm4.25 765	Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
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Theorem	orim12i 766	Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
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Theorem	orim1i 767	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
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Theorem	orim2i 768	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
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Theorem	orbi2i 769	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.)
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Theorem	orbi1i 770	Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theorem	orbi12i 771	Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
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Theorem	pm1.5 772	Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
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Theorem	or12 773	Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
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Theorem	orass 774	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.)
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Theorem	pm2.31 775	Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.32 776	Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
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Theorem	or32 777	A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem	or4 778	Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
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Theorem	or42 779	Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
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Theorem	orordi 780	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
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Theorem	orordir 781	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
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Theorem	pm2.3 782	Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.41 783	Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.42 784	Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.4 785	Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm4.44 786	Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm4.56 787	Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
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Theorem	oranim 788	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.)
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Theorem	pm4.78i 789	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.)
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Theorem	mtord 790	A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.)
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Theorem	pm4.45 791	Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm3.48 792	Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.)
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Theorem	orim12d 793	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
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Theorem	orim1d 794	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
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Theorem	orim2d 795	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
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Theorem	orim2 796	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
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Theorem	orbi2d 797	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.)
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Theorem	orbi1d 798	Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theorem	orbi1 799	Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
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Theorem	orbi12d 800	Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)
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