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**Theorem List for Intuitionistic Logic Explorer -** 601-700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	pm2.01da 601	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
$/\$

Theorem	pm3.2im 602	In classical logic, this is just a restatement of pm3.2 138. In intuitionistic logic, it still holds, but is weaker than pm3.2. (Contributed by Mario Carneiro, 12-May-2015.)


Theorem	expi 603	An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)


Theorem	pm2.65i 604	Inference for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)


Theorem	mt2 605	A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)


Theorem	biijust 606	Theorem used to justify definition of intuitionistic biconditional df-bi 116. (Contributed by NM, 24-Nov-2017.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	con3 607	Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)


Theorem	con2 608	Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)


Theorem	mt2i 609	Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)


Theorem	notnoti 610	Infer double negation. (Contributed by NM, 27-Feb-2008.)


Theorem	pm2.21i 611	A contradiction implies anything. Inference from pm2.21 583. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm2.24ii 612	A contradiction implies anything. Inference from pm2.24 587. (Contributed by NM, 27-Feb-2008.)


Theorem	nsyld 613	A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)


Theorem	nsyli 614	A negated syllogism inference. (Contributed by NM, 3-May-1994.)


Theorem	mth8 615	Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)


Theorem	jc 616	Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)


Theorem	pm2.51 617	Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)


Theorem	pm2.52 618	Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	expt 619	Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)


Theorem	jarl 620	Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)


Theorem	pm2.65 621	Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here , derive a contradiction, and therefore conclude , are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume , derive a contradiction, and conclude , such as condandc 814, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)


Theorem	pm2.65d 622	Deduction for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)


Theorem	pm2.65da 623	Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
$/\$ $/\$

Theorem	mto 624	The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)


Theorem	mtod 625	Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)


Theorem	mtoi 626	Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)


Theorem	mtand 627	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
$/\$

Theorem	notbid 628	Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	con2b 629	Contraposition. Bidirectional version of con2 608. (Contributed by NM, 5-Aug-1993.)


Theorem	notbii 630	Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	mtbi 631	An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)


Theorem	mtbir 632	An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)


Theorem	mtbid 633	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)


Theorem	mtbird 634	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)


Theorem	mtbii 635	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)


Theorem	mtbiri 636	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)


Theorem	sylnib 637	A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)


Theorem	sylnibr 638	A mixed syllogism inference from an implication and a biconditional. Useful for substituting an consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)


Theorem	sylnbi 639	A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)


Theorem	sylnbir 640	A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)


Theorem	xchnxbi 641	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)


Theorem	xchnxbir 642	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)


Theorem	xchbinx 643	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)


Theorem	xchbinxr 644	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)


Theorem	mt2bi 645	A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)


Theorem	mtt 646	Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm5.21 647	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.)
$/\$

Theorem	pm5.21im 648	Two propositions are equivalent if they are both false. Closed form of 2false 653. Equivalent to a bi2 129-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	nbn2 649	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 650	Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)


Theorem	nbn 651	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 652	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)


Theorem	2false 653	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	2falsed 654	Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)


Theorem	pm5.21ni 655	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 656	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 657	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	pm5.19 658	Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm4.8 659	Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 853 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)


Theorem	imnan 660	Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
$/\$

Theorem	imnani 661	Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
$/\$

Theorem	nan 662	Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
$/\$ $/\$

Theorem	pm3.24 663	Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 2-Feb-2015.)
$/\$

Theorem	notnotnot 664	Triple negation. (Contributed by Jim Kingdon, 28-Jul-2018.)


1.2.6 Logical disjunction

Syntax	wo 665	Extend wff definition to include disjunction ('or').
$\/$

Axiom	ax-io 666	Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 667 instead. (New usage is discouraged.)
$\/$ $/\$

Theorem	jaob 667	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 666. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$ $/\$

Theorem	olc 668	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
$\/$

Theorem	orc 669	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
$\/$

Theorem	pm2.67-2 670	Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
$\/$

Theorem	pm3.44 671	Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
$/\$ $\/$

Theorem	jaoi 672	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
$\/$

Theorem	jaod 673	Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.)
$\/$

Theorem	mpjaod 674	Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
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Theorem	jaao 675	Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
$/\$ $\/$

Theorem	jaoa 676	Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
$\/$ $/\$

Theorem	pm2.53 677	Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 829). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
$\/$

Theorem	ori 678	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

Theorem	ord 679	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

Theorem	orel1 680	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.)
$\/$

Theorem	orel2 681	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.)
$\/$

Theorem	pm1.4 682	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
$\/$ $\/$

Theorem	orcom 683	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.)
$\/$ $\/$

Theorem	orcomd 684	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
$\/$ $\/$

Theorem	orcoms 685	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
$\/$ $\/$

Theorem	orci 686	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

Theorem	olci 687	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

Theorem	orcd 688	Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
$\/$

Theorem	olcd 689	Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
$\/$

Theorem	orcs 690	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 691	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
$\/$

Theorem	pm2.07 692	Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
$\/$

Theorem	pm2.45 693	Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
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Theorem	pm2.46 694	Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
$\/$

Theorem	pm2.47 695	Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
$\/$ $\/$

Theorem	pm2.48 696	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
$\/$ $\/$

Theorem	pm2.49 697	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
$\/$ $\/$

Theorem	pm2.67 698	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 699	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 700	A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
$\/$

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