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

Theorem	mt2 601	A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)


Theorem	biijust 602	Theorem used to justify definition of intuitionistic biconditional df-bi 115. (Contributed by NM, 24-Nov-2017.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	con3 603	Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)


Theorem	con2 604	Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)


Theorem	mt2i 605	Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)


Theorem	notnoti 606	Infer double negation. (Contributed by NM, 27-Feb-2008.)


Theorem	pm2.21i 607	A contradiction implies anything. Inference from pm2.21 579. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm2.24ii 608	A contradiction implies anything. Inference from pm2.24 583. (Contributed by NM, 27-Feb-2008.)


Theorem	nsyld 609	A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)


Theorem	nsyli 610	A negated syllogism inference. (Contributed by NM, 3-May-1994.)


Theorem	mth8 611	Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)


Theorem	jc 612	Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)


Theorem	pm2.51 613	Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)


Theorem	pm2.52 614	Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	expt 615	Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)


Theorem	jarl 616	Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)


Theorem	pm2.65 617	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 808, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)


Theorem	pm2.65d 618	Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)


Theorem	pm2.65da 619	Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
$/\$ $/\$

Theorem	mto 620	The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)


Theorem	mtod 621	Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)


Theorem	mtoi 622	Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)


Theorem	mtand 623	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
$/\$

Theorem	notbid 624	Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	con2b 625	Contraposition. Bidirectional version of con2 604. (Contributed by NM, 5-Aug-1993.)


Theorem	notbii 626	Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	mtbi 627	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 628	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 629	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)


Theorem	mtbird 630	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)


Theorem	mtbii 631	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)


Theorem	mtbiri 632	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)


Theorem	sylnib 633	A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)


Theorem	sylnibr 634	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 635	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 636	A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)


Theorem	xchnxbi 637	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)


Theorem	xchnxbir 638	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)


Theorem	xchbinx 639	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)


Theorem	xchbinxr 640	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)


Theorem	mt2bi 641	A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)


Theorem	mtt 642	Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm5.21 643	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 644	Two propositions are equivalent if they are both false. Closed form of 2false 649. Equivalent to a bi2 128-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	nbn2 645	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 646	Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)


Theorem	nbn 647	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 648	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)


Theorem	2false 649	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	2falsed 650	Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)


Theorem	pm5.21ni 651	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 652	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 653	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	pm5.19 654	Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)


Theorem	pm4.8 655	Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 847 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)


Theorem	imnan 656	Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
$/\$

Theorem	imnani 657	Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
$/\$

Theorem	nan 658	Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
$/\$ $/\$

Theorem	pm3.24 659	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 660	Triple negation. (Contributed by Jim Kingdon, 28-Jul-2018.)


1.2.6 Logical disjunction

Syntax	wo 661	Extend wff definition to include disjunction ('or').
$\/$

Axiom	ax-io 662	Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 663 instead. (New usage is discouraged.)
$\/$ $/\$

Theorem	jaob 663	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 662. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$ $/\$

Theorem	olc 664	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 665	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
$\/$

Theorem	pm2.67-2 666	Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
$\/$

Theorem	pm3.44 667	Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
$/\$ $\/$

Theorem	jaoi 668	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
$\/$

Theorem	jaod 669	Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.)
$\/$

Theorem	mpjaod 670	Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
$\/$

Theorem	jaao 671	Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
$/\$ $\/$

Theorem	jaoa 672	Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
$\/$ $/\$

Theorem	pm2.53 673	Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 823). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
$\/$

Theorem	ori 674	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

Theorem	ord 675	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

Theorem	orel1 676	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 677	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	pm1.4 678	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
$\/$ $\/$

Theorem	orcom 679	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 680	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
$\/$ $\/$

Theorem	orcoms 681	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
$\/$ $\/$

Theorem	orci 682	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

Theorem	olci 683	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

Theorem	orcd 684	Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
$\/$

Theorem	olcd 685	Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
$\/$

Theorem	orcs 686	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.)
$\/$

Theorem	olcs 687	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
$\/$

Theorem	pm2.07 688	Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
$\/$

Theorem	pm2.45 689	Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
$\/$

Theorem	pm2.46 690	Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
$\/$

Theorem	pm2.47 691	Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
$\/$ $\/$

Theorem	pm2.48 692	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
$\/$ $\/$

Theorem	pm2.49 693	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
$\/$ $\/$

Theorem	pm2.67 694	Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
$\/$

Theorem	biorf 695	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.)
$\/$

Theorem	biortn 696	A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
$\/$

Theorem	biorfi 697	A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
$\/$

Theorem	pm2.621 698	Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
$\/$

Theorem	pm2.62 699	Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
$\/$

Theorem	imorri 700	Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
$\/$

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