P. 6 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 501-600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	an13s 501	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑)) → 𝜃)
Theorem	an32s 502	Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃)
Theorem	ancom1s 503	Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) → 𝜃)
Theorem	an31s 504	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜒 ∧ 𝜓) ∧ 𝜑) → 𝜃)
Theorem	anass1rs 505	Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃)
Theorem	anabs1 506	Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) ↔ (𝜑 ∧ 𝜓))
Theorem	anabs5 507	Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
Theorem	anabs7 508	Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
Theorem	anabsan 509	Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss1 510	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss4 511	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
⊢ (((𝜓 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss5 512	Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi5 513	Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi6 514	Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
⊢ (𝜑 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi7 515	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
⊢ (𝜓 → ((𝜑 ∧ 𝜓) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi8 516	Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
⊢ (𝜓 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss7 517	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsan2 518	Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss3 519	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	an4 520	Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
Theorem	an42 521	Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
Theorem	an4s 522	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) → 𝜏)
Theorem	an42s 523	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) → 𝜏)
Theorem	anandi 524	Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandir 525	Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)))
Theorem	anandis 526	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏)
Theorem	anandirs 527	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜏)
Theorem	impbida 528	Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm3.45 529	Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
Theorem	im2anan9 530	Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))
Theorem	im2anan9r 531	Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))
Theorem	anim12dan 532	Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → (𝜒 ∧ 𝜏))
Theorem	pm5.1 533	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	pm3.43 534	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 535	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	pm4.76 536	Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	pm4.38 537	Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	bi2anan9 538	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 539	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 540	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	pm5.33 541	Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒)))
Theorem	pm5.36 542	Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) ↔ (𝜓 ∧ (𝜑 ↔ 𝜓)))
Theorem	bianabs 543	Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
1.2.5  Logical negation (intuitionistic)
Axiom	ax-in1 544	'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Axiom	ax-in2 545	'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (¬ 𝜑 → (𝜑 → 𝜓))
Theorem	pm2.01 546	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 750 which only holds for some propositions. (Contributed by Mario Carneiro, 12-May-2015.)
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑)
Theorem	pm2.21 547	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 548	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 549	A contradiction implies anything. Deduction from pm2.21 547. (Contributed by NM, 10-Feb-1996.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	pm2.21dd 550	A contradiction implies anything. Deduction from pm2.21 547. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	pm2.24 551	Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
⊢ (𝜑 → (¬ 𝜑 → 𝜓))
Theorem	pm2.24d 552	Deduction version of pm2.24 551. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
Theorem	pm2.24i 553	Inference version of pm2.24 551. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜑 → 𝜓)
Theorem	con2d 554	A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 → ¬ 𝜓))
Theorem	mt2d 555	Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → (𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	nsyl3 556	A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ (𝜒 → ¬ 𝜑)
Theorem	con2i 557	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 558	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 559	Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101. This one holds for all propositions, but its converse only holds for decidable propositions (see notnotrdc 751). (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
⊢ (𝜑 → ¬ ¬ 𝜑)
Theorem	notnotd 560	Deduction associated with notnot 559 and notnoti 574. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ¬ ¬ 𝜓)
Theorem	con3d 561	A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
Theorem	con3i 562	A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜓 → ¬ 𝜑)
Theorem	con3rr3 563	Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (¬ 𝜒 → (𝜑 → ¬ 𝜓))
Theorem	con3and 564	Variant of con3d 561 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)
Theorem	pm2.01da 565	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	pm3.2im 566	In classical logic, this is just a restatement of pm3.2 126. 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 567	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 568	Inference rule 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 569	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 570	Theorem used to justify definition of intuitionistic biconditional df-bi 110. (Contributed by NM, 24-Nov-2017.)
⊢ ((((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
Theorem	con3 571	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 572	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 573	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 574	Infer double negation. (Contributed by NM, 27-Feb-2008.)
⊢ 𝜑    ⇒   ⊢ ¬ ¬ 𝜑
Theorem	pm2.21i 575	A contradiction implies anything. Inference from pm2.21 547. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.24ii 576	A contradiction implies anything. Inference from pm2.24 551. (Contributed by NM, 27-Feb-2008.)
⊢ 𝜑    &   ⊢ ¬ 𝜑    ⇒   ⊢ 𝜓
Theorem	nsyld 577	A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒))    &   ⊢ (𝜑 → (𝜏 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ¬ 𝜏))
Theorem	nsyli 578	A negated syllogism inference. (Contributed by NM, 3-May-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → ¬ 𝜒)    ⇒   ⊢ (𝜑 → (𝜃 → ¬ 𝜓))
Theorem	mth8 579	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	jc 580	Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ¬ (𝜓 → ¬ 𝜒))
Theorem	pm2.51 581	Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ (𝜑 → 𝜓) → (𝜑 → ¬ 𝜓))
Theorem	pm2.52 582	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 583	Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
⊢ ((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))
Theorem	jarl 584	Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (¬ 𝜑 → 𝜒))
Theorem	pm2.65 585	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 775, 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 586	Deduction rule 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 587	Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mto 588	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 589	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 590	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 591	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	notbid 592	Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
Theorem	con2b 593	Contraposition. Bidirectional version of con2 572. (Contributed by NM, 5-Aug-1993.)
⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
Theorem	notbii 594	Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
Theorem	mtbi 595	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 596	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 597	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
⊢ (𝜑 → ¬ 𝜓)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbird 598	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
⊢ (𝜑 → ¬ 𝜒)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	mtbii 599	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
⊢ ¬ 𝜓    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	mtbiri 600	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
⊢ ¬ 𝜒    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)