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	bi2anan9 601	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 602	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 603	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	pm5.33 604	Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒)))
Theorem	pm5.36 605	Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) ↔ (𝜓 ∧ (𝜑 ↔ 𝜓)))
Theorem	bianabs 606	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 607	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 608	Inference associated with biadani 607. 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 609	'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias pm2.01 611 instead. (New usage is discouraged.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Axiom	ax-in2 610	'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (¬ 𝜑 → (𝜑 → 𝜓))
Theorem	pm2.01 611	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 850 which only holds for some propositions. Also called weak Clavius law. (Contributed by Mario Carneiro, 12-May-2015.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Theorem	pm2.21 612	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 613	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 614	A contradiction implies anything. Deduction from pm2.21 612. (Contributed by NM, 10-Feb-1996.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	pm2.21dd 615	A contradiction implies anything. Deduction from pm2.21 612. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	pm2.24 616	Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → (¬ 𝜑 → 𝜓))
Theorem	pm2.24d 617	Deduction version of pm2.24 616. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
Theorem	pm2.24i 618	Inference version of pm2.24 616. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜑 → 𝜓)
Theorem	con2d 619	A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 → ¬ 𝜓))
Theorem	mt2d 620	Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → (𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	nsyl3 621	A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ (𝜒 → ¬ 𝜑)
Theorem	con2i 622	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 623	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 624	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 826) and in particular for decidable propositions (see notnotrdc 838). See also notnotnot 629. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
⊢ (𝜑 → ¬ ¬ 𝜑)
Theorem	notnotd 625	Deduction associated with notnot 624 and notnoti 640. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ¬ ¬ 𝜓)
Theorem	con3d 626	A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
Theorem	con3i 627	A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜓 → ¬ 𝜑)
Theorem	con3rr3 628	Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (¬ 𝜒 → (𝜑 → ¬ 𝜓))
Theorem	notnotnot 629	Triple negation is equivalent to negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
⊢ (¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)
Theorem	con3dimp 630	Variant of con3d 626 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)
Theorem	pm2.01da 631	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm3.2im 632	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.)
⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
Theorem	expi 633	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 634	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 635	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 636	Theorem used to justify definition of intuitionistic biconditional df-bi 116. (Contributed by NM, 24-Nov-2017.)
⊢ ((((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
Theorem	con3 637	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 638	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 639	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 640	Infer double negation. (Contributed by NM, 27-Feb-2008.)
⊢ 𝜑    ⇒   ⊢ ¬ ¬ 𝜑
Theorem	pm2.21i 641	A contradiction implies anything. Inference from pm2.21 612. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.24ii 642	A contradiction implies anything. Inference from pm2.24 616. (Contributed by NM, 27-Feb-2008.)
⊢ 𝜑    &   ⊢ ¬ 𝜑    ⇒   ⊢ 𝜓
Theorem	nsyld 643	A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒))    &   ⊢ (𝜑 → (𝜏 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ¬ 𝜏))
Theorem	nsyli 644	A negated syllogism inference. (Contributed by NM, 3-May-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → (𝜃 → ¬ 𝜓))
Theorem	jc 645	Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ¬ (𝜓 → ¬ 𝜒))
Theorem	jcn 646	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 647	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 648	Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
⊢ (¬ (𝜑 → 𝜓) → ¬ 𝜓)
Theorem	conax1k 649	Weakening of conax1 648. General instance of pm2.51 650 and of pm2.52 651. (Contributed by BJ, 28-Oct-2023.)
⊢ (¬ (𝜑 → 𝜓) → (𝜒 → ¬ 𝜓))
Theorem	pm2.51 650	Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 → 𝜓) → (𝜑 → ¬ 𝜓))
Theorem	pm2.52 651	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 652	Exportation theorem pm3.3 259 (closed form of ex 114) expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
⊢ ((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))
Theorem	jarl 653	Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (¬ 𝜑 → 𝜒))
Theorem	pm2.65 654	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 876, 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 655	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 656	Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mto 657	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 658	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 659	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 660	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	notbi 661	Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)
⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
Theorem	notbid 662	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 663	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 664	Contraposition. Bidirectional version of con2 638. (Contributed by NM, 5-Aug-1993.)
⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
Theorem	mtbi 665	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 666	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 667	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbird 668	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mtbii 669	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
⊢ ¬ 𝜓    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbiri 670	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
⊢ ¬ 𝜒    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	sylnib 671	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 672	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 673	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 674	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 675	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (¬ 𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    ⇒   ⊢ (¬ 𝜒 ↔ 𝜓)
Theorem	xchnxbir 676	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (¬ 𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    ⇒   ⊢ (¬ 𝜒 ↔ 𝜓)
Theorem	xchbinx 677	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜒)
Theorem	xchbinxr 678	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜒)
Theorem	mt2bi 679	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 680	Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑)))
Theorem	annimim 681	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 932. (Contributed by Jim Kingdon, 24-Dec-2017.)
⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
Theorem	pm4.65r 682	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 683	Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 884. (Contributed by Jim Kingdon, 24-Dec-2017.)
⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
Theorem	pm3.37 684	Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ∧ 𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
Theorem	imnan 685	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 686	Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
⊢ ¬ (𝜑 ∧ 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	nan 687	Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))
Theorem	pm3.24 688	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	pm4.15 689	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	pm5.21 690	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 691	Two propositions are equivalent if they are both false. Closed form of 2false 696. Equivalent to a biimpr 129-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 692	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 693	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 694	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 695	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 ↔ (𝜓 ↔ ¬ 𝜑))
Theorem	2false 696	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ ¬ 𝜑    &   ⊢ ¬ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	2falsed 697	Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.21ni 698	Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ (¬ 𝜓 → (𝜑 ↔ 𝜒))
Theorem	pm5.21nii 699	Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜓)    &   ⊢ (𝜓 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	pm5.21ndd 700	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.)
⊢ (𝜑 → (𝜒 → 𝜓))    &   ⊢ (𝜑 → (𝜃 → 𝜓))    &   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜃))