P. 7 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	pm5.1 601	Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓))
Theorem	pm3.43 602	Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 27-Nov-2013.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	jcab 603	Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))
Theorem	pm4.76 604	Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	pm4.38 605	Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	bi2anan9 606	Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜏 ↔ 𝜂))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))
Theorem	bi2anan9r 607	Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜏 ↔ 𝜂))    ⇒   ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))
Theorem	bi2bian9 608	Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜏 ↔ 𝜂))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ↔ 𝜏) ↔ (𝜒 ↔ 𝜂)))
Theorem	pm5.33 609	Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒)))
Theorem	pm5.36 610	Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) ↔ (𝜓 ∧ (𝜑 ↔ 𝜓)))
Theorem	bianabs 611	Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	biadani 612	An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (Contributed by BJ, 4-Mar-2023.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	biadanii 613	Inference associated with biadani 612. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
1.2.5  Logical negation (intuitionistic)
Axiom	ax-in1 614	'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias pm2.01 616 instead. (New usage is discouraged.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Axiom	ax-in2 615	'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (¬ 𝜑 → (𝜑 → 𝜓))
Theorem	pm2.01 616	Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. This is valid intuitionistically (in the terminology of Section 1.2 of [Bauer] p. 482 it is a proof of negation not a proof by contradiction); compare with pm2.18dc 855 which only holds for some propositions. Also called weak Clavius law. (Contributed by Mario Carneiro, 12-May-2015.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Theorem	pm2.21 617	From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. (Contributed by Mario Carneiro, 12-May-2015.)
⊢ (¬ 𝜑 → (𝜑 → 𝜓))
Theorem	pm2.01d 618	Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 → (𝜓 → ¬ 𝜓))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm2.21d 619	A contradiction implies anything. Deduction from pm2.21 617. (Contributed by NM, 10-Feb-1996.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	pm2.21dd 620	A contradiction implies anything. Deduction from pm2.21 617. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	pm2.24 621	Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → (¬ 𝜑 → 𝜓))
Theorem	pm2.24d 622	Deduction version of pm2.24 621. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
Theorem	pm2.24i 623	Inference version of pm2.24 621. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜑 → 𝜓)
Theorem	con2d 624	A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 → ¬ 𝜓))
Theorem	mt2d 625	Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → (𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	nsyl3 626	A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ (𝜒 → ¬ 𝜑)
Theorem	con2i 627	A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜓 → ¬ 𝜑)
Theorem	nsyl 628	A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	notnot 629	Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101. The converse need not hold. It holds exactly for stable propositions (by definition, see df-stab 831) and in particular for decidable propositions (see notnotrdc 843). See also notnotnot 634. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
⊢ (𝜑 → ¬ ¬ 𝜑)
Theorem	notnotd 630	Deduction associated with notnot 629 and notnoti 645. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ¬ ¬ 𝜓)
Theorem	con3d 631	A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
Theorem	con3i 632	A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜓 → ¬ 𝜑)
Theorem	con3rr3 633	Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (¬ 𝜒 → (𝜑 → ¬ 𝜓))
Theorem	notnotnot 634	Triple negation is equivalent to negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
⊢ (¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)
Theorem	con3dimp 635	Variant of con3d 631 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)
Theorem	pm2.01da 636	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm3.2im 637	In classical logic, this is just a restatement of pm3.2 139. In intuitionistic logic, it still holds, but is weaker than pm3.2. (Contributed by Mario Carneiro, 12-May-2015.)
⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
Theorem	expi 638	An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
⊢ (¬ (𝜑 → ¬ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	pm2.65i 639	Inference for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ ¬ 𝜑
Theorem	mt2 640	A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
⊢ 𝜓    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ ¬ 𝜑
Theorem	biijust 641	Theorem used to justify definition of intuitionistic biconditional df-bi 117. (Contributed by NM, 24-Nov-2017.)
⊢ ((((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
Theorem	con3 642	Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
Theorem	con2 643	Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
⊢ ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
Theorem	mt2i 644	Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
⊢ 𝜒    &   ⊢ (𝜑 → (𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	notnoti 645	Infer double negation. (Contributed by NM, 27-Feb-2008.)
⊢ 𝜑    ⇒   ⊢ ¬ ¬ 𝜑
Theorem	pm2.21i 646	A contradiction implies anything. Inference from pm2.21 617. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.24ii 647	A contradiction implies anything. Inference from pm2.24 621. (Contributed by NM, 27-Feb-2008.)
⊢ 𝜑    &   ⊢ ¬ 𝜑    ⇒   ⊢ 𝜓
Theorem	nsyld 648	A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒))    &   ⊢ (𝜑 → (𝜏 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ¬ 𝜏))
Theorem	nsyli 649	A negated syllogism inference. (Contributed by NM, 3-May-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → (𝜃 → ¬ 𝜓))
Theorem	jc 650	Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ¬ (𝜓 → ¬ 𝜒))
Theorem	jcn 651	Theorem joining the consequents of two premises. Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
⊢ (𝜑 → (¬ 𝜓 → ¬ (𝜑 → 𝜓)))
Theorem	jcnd 652	Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → ¬ (𝜓 → 𝜒))
Theorem	conax1 653	Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
⊢ (¬ (𝜑 → 𝜓) → ¬ 𝜓)
Theorem	conax1k 654	Weakening of conax1 653. General instance of pm2.51 655 and of pm2.52 656. (Contributed by BJ, 28-Oct-2023.)
⊢ (¬ (𝜑 → 𝜓) → (𝜒 → ¬ 𝜓))
Theorem	pm2.51 655	Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 → 𝜓) → (𝜑 → ¬ 𝜓))
Theorem	pm2.52 656	Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (¬ (𝜑 → 𝜓) → (¬ 𝜑 → ¬ 𝜓))
Theorem	expt 657	Exportation theorem pm3.3 261 (closed form of ex 115) expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
⊢ ((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))
Theorem	jarl 658	Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (¬ 𝜑 → 𝜒))
Theorem	pm2.65 659	Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here 𝜑, derive a contradiction, and therefore conclude ¬ 𝜑, 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 ¬ 𝜑, derive a contradiction, and conclude 𝜑, such as condandc 881, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))
Theorem	pm2.65d 660	Deduction for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm2.65da 661	Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mto 662	The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
⊢ ¬ 𝜓    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ ¬ 𝜑
Theorem	mtod 663	Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mtoi 664	Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
⊢ ¬ 𝜒    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mtand 665	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	notbi 666	Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)
⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
Theorem	notbid 667	Equivalence property for negation. Deduction form. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
Theorem	notbii 668	Equivalence property for negation. Inference form. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
Theorem	con2b 669	Contraposition. Bidirectional version of con2 643. (Contributed by NM, 5-Aug-1993.)
⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
Theorem	mtbi 670	An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ ¬ 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ¬ 𝜓
Theorem	mtbir 671	An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
⊢ ¬ 𝜓    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ¬ 𝜑
Theorem	mtbid 672	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbird 673	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mtbii 674	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
⊢ ¬ 𝜓    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbiri 675	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
⊢ ¬ 𝜒    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	sylnib 676	A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	sylnibr 677	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.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	sylnbi 678	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.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (¬ 𝜓 → 𝜒)    ⇒   ⊢ (¬ 𝜑 → 𝜒)
Theorem	sylnbir 679	A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (¬ 𝜓 → 𝜒)    ⇒   ⊢ (¬ 𝜑 → 𝜒)
Theorem	xchnxbi 680	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (¬ 𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    ⇒   ⊢ (¬ 𝜒 ↔ 𝜓)
Theorem	xchnxbir 681	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (¬ 𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    ⇒   ⊢ (¬ 𝜒 ↔ 𝜓)
Theorem	xchbinx 682	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜒)
Theorem	xchbinxr 683	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜒)
Theorem	mt2bi 684	A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 ↔ (𝜓 → ¬ 𝜑))
Theorem	mtt 685	Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑)))
Theorem	annimim 686	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 937. (Contributed by Jim Kingdon, 24-Dec-2017.)
⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
Theorem	pm4.65r 687	One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓))
Theorem	imanim 688	Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 889. (Contributed by Jim Kingdon, 24-Dec-2017.)
⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
Theorem	pm3.37 689	Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∧ 𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
Theorem	imnan 690	Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
Theorem	imnani 691	Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
⊢ ¬ (𝜑 ∧ 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	nan 692	Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))
Theorem	pm3.24 693	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.)
⊢ ¬ (𝜑 ∧ ¬ 𝜑)
Theorem	pm4.15 694	Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
⊢ (((𝜑 ∧ 𝜓) → ¬ 𝜒) ↔ ((𝜓 ∧ 𝜒) → ¬ 𝜑))
Theorem	pm5.21 695	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.)
⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓))
Theorem	pm5.21im 696	Two propositions are equivalent if they are both false. Closed form of 2false 701. 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.)
⊢ (¬ 𝜑 → (¬ 𝜓 → (𝜑 ↔ 𝜓)))
Theorem	nbn2 697	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.)
⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))
Theorem	bibif 698	Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
⊢ (¬ 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ 𝜑))
Theorem	nbn 699	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.)
⊢ ¬ 𝜑    ⇒   ⊢ (¬ 𝜓 ↔ (𝜓 ↔ 𝜑))
Theorem	nbn3 700	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 ↔ (𝜓 ↔ ¬ 𝜑))